An /ɝ/ is just the stressed version of an /ɚ/. For example, murder has both of them in it, being normally written as /ˈmɝdɚ/. Both of those are “r-colored” vowels. However, some transcribers prefer to represent that as /ˈmɜɹdəɹ/ instead, writing a consonant instead of little rhotic hook. Those represent the same pronunciation.

Your mother is therefore going to be either your /ˈmʌðɚ/ or your /ˈmʌðəɹ/. You need to understand though that /ɚ/ and /əɹ/ are just two ways of writing the same thing — at least in words like murder and mother. When you can get into words like murdering or mothering, then you cannot use the r-colored version for the one before the -ing, since it now has a vowel after it and so much be written as a consonant.

There are advantages and disadvantages to doing it one way or the other. Using a consonant instead of a diacritic can be easier to understand, since you don’t have to think about whether it has a consonant following it and so counts as a rhotacized vowel, or whether it has a vowel following it and so counts as a consonant.

• mirth:          /ˈmɝθ/      or  /ˈmɜɹθ/
• mother:       /ˈmʌðɚ/      or  /ˈmʌðəɹ/
• mothering: /ˈmʌðəɹɪŋ/     
• murder:       /ˈmɝdɚ/      or  /ˈmɜɹdəɹ/
• murdered:   /ˈmɝdɚd/    or  /ˈmɜɹdəɹd/
• murderous: /ˈmɝdəɹəs/ or  /ˈmɜɹdəɹəs/
• murderer:    /ˈmɝdɚɚ/    or  /ˈmɜɹdəɹəɹ/

Another issue is that IPA doesn’t have special precomposed characters for other rhotacized vowels, so you have to build the others yourself, which means they don’t look like the precomposed ones:

• Mordor:               /ˈmo˞do˞/ or  /ˈmoɹdoɹ/
• corner:                 /ˈko˞nɚ/   or  /ˈkoɹnəɹ/
• harder:                /ˈhɑ˞dɚ/    or  /ˈhɑɹdəɹ/
• radar:                 /ˈɹeɪdɑ˞/   or  /ˈɹeɪdɑɹ/
• carport:              /ˈkɑ˞po˞t/ or  /ˈkɑɹpoɹt/
• rarer:                  /ˈɹeɪɹɚ/    or  /ˈɹeɪɹəɹ/
• creature:             /ˈkɹitʃɚ/  or  /ˈkɹitʃəɹ/
• entrepreneur:     /ˌɔntɹəpɹəˈnɚ/ or /ˌɔntɹəpɹəˈnɜɹ/
• entrepreneurial: /ˌɔntɹəpɹəˈnʊɹiəl/

If you are doing phonemic transcriptions, you might consider just sticking with /r/ and not worrying about all the various phonetic realizations possible for it, including [ɝ], [ɚ], [ɹ], [ɻʷ], and all the rest.

As for trying to identify difference between the several Canadian accents and the many American ones, when it comes to your r’s, this mostly depends on whether you are comparing rhotic dialects with non-rhotic ones. Note also that the standard versions of both sets are rhotic. That means your mother is still going to be the same wherever you are, and that you don’t need to worry about it. The mother of Vancouver is the same as the one from San José.

You haven’t said whether your first language is some variety of English, or whether it is something else. If it is something else, especially one without the sorts of rhotics that occur in North America, then simply mastering those alone will be much harder, and much more important, than trying to tease out one or another difference between this or that American or Canadian accent.

There are a large number of words pronounced with a long vowel before -ic. However, there are some rules that can help predict when a word will be an exception to the shortening rule.

For many of the less predictable words, both pronunciations are recorded in dictionaries.

Sources I've found that mention both the shortening rule and some exceptions to it: English Word-Stress, by Erik Fudge, and A Survey of English Spelling, by Edward Carney.

Vowels directly before -ic are generally long

First of all, a long vowel is used in words where the stressed vowel directly precedes the suffix with no intervening consonant, as in algebraic, nucleic, benzoic.

The only tricky part here is that some words ending in eic are pronounced by some speakers with long a (/ˈeɪɪk/) rather than long e (/ˈiːɪk/). See for example the pronunciation Collins gives for allogeneic. This seems comparable to the change of from /ˈiːɪti/ to /ˈeɪɪti/ that has occurred for some speakers in words like deity and spontaneity.

Some vowels or digraphs don't have short pronunciations

Long vowels are also used before -ic in words written with vowels or digraphs that don't have alternative short pronunciations (at least, not in this context) such as au, u, ou, eu, or ei. The digraphs au, eu and ei, and usually ou, are also immune to closed-syllable shortening (consider the pronunciation of ei as /aɪ/ in deictic and seismic). Here are some examples:

With au*:     With u/ou:       With eu:             With ei:
caustic      cherubic         therapeutic           oneiric
hydraulic    uric             hermeneutic           isocheimic         
aeronautic   cupric           pharmeceutic(al)     
banausic     anacoluthic                           
             abulic/aboulic
             acoustic**

*Erik Fudge says caustic and hydraulic may have /ɒ/, but I'm not sure if that would be due to pre-ic shortening, or to a more general process that sometimes shortens /ɔː/ to /ɒ/ in British English, especially before consonant clusters—see this question for more details: How do you pronounce "bald"?
**Erik Fudge says "acoustic" may have /ʊ/, but I haven't found any other source tha corroborates this.

There are some exceptions to the generalization that u is long in an orthographically open syllable before -ic. The word public and other words that end with -public do have short u (IPA /ʌ/), as does rustic. The word bismuthic, which is derived from the word bismuth with a short u /ʌ/, may be pronounced with either a short /ʌ/ or long /juː/. Foreign words also don't always follow the same rules. The word Talmudic is often pronounced with long /juː/ or /uː/, but it can also be pronounced with short /ʊ/ or /ʌ/, using the same vowel as in the word Talmud. Speaking of foreign words, that's our next category.

Long vowels in foreign words may be unchanged before -ic

A long vowel may also occur before -ic in adjectives formed from words of foreign origin that have not been fully adapted to the English sound/spelling system.

For example, foreign words spelled with a, such as Islam, Koran (also spelled Quran or Qur'an), and Purana, can often be pronounced with either short a (/æ/) or broad a (/ɑː/). Broad /ɑː/ doesn't have a short equivalent within the vowel alternation system of English, and it is not changed by the addition of the suffix -ic. I'm not sure if there are any people who use broad /ɑː/ in "Islam" and short /æ/ in "Islamic," or vice versa. I would guess that most people are consistent and use the same vowel in both the noun and the corresponding adjective.

In some words that come from or were influenced by French, the letter i can be pronounced as tense e (that is, /iː/). For example, nicotine always has tense /iː/, amino may have tense /iː/, and bulimia may have tense /iː/. Either this tense vowel /iː/ or the English short i vowel /ɪ/ may be used in the related adjectives.

With /æ/ or /ɑː/:     With /ɪ/ or /iː/: 
Islamic               nicotinic 
Koranic               aminic 
Puranic               bulimic
Slavic
halachic/halachic
midrashic
akashic
saccadic (oddly, Collins also lists a pronunciation with /eɪ/)

There are a handful of words like this with ith long e/a or o:

Always tense:
Vedic       (from Veda; either long a /eɪ/ or long e /iː/)
Yogic/yogic (from yoga; has long o /oʊ/)
ohmic       (from ohm; has long o /oʊ/)

Sometimes tense:
Gaelic      (has either long a /eɪ/ or short a /æ/)
stearic     (has either short /iːˈæ/ or long e /iː/)

Ohmic is unusual, but arguably, the spelling pattern "oh" cannot represent a short vowel in English (it's similarly immune to closed-syllable laxing in the word ohm).

Gaelic has an odd spelling pattern where "ae" = /eɪ/ for most speakers, although it may also be pronounced with /æ/ (or even, according to Merriam Webster, with broad /ɑː/).

Stearic comes from a French word with two separate vowels. It may be pronounced with separate vowels in English also (/stiˈærik/), and some dictionaries only list that pronunciation, but several dictionaries additionally list a pronunciation with the ea as the long vowel found in fear: American Heritage Dictionary, Dictionary.com.

Ae and oe are often long, even if the word is now spelled with e

Another class of exceptions is words that are, or that used to be, spelled with a digraph ae or oe. Many of these are now just spelled with e, but there seem to be some lingering pronunciation differences:

Always tense:     Sometimes tense:
blastocoelic      chim(a)eric  
am(o)ebic
an(a)emic
encyclop(a)edic
orthop(a)edic

In fact, this includes all medical terms ending in -(a)emic that are derived from the root haem "blood", such as glyc(a)emic, emphysemic, empyemic, antileukemic, agammaglobulinemic. Most of these have corresponding nouns ending in -emia.

Irregular specific words and endings that do not undergo laxing

The other exceptional words contain vowels that normally would be expected to undergo laxing, and that have spelling consistent with a lax pronunciation, but that for whatever reason can or must be pronounced with long vowels. In general, the cause of this seems to be influence from related words where the vowel is kong.

There are certain word-endings containing the suffix -ic that invariably have long vowels. The ending -phobic always has long o; it is related to the endings -phobia and -phobe, which both have long o. The suffix -emic always has long e; it is related to the suffix -eme, which has long e. This suffix is derived from the word phoneme, and is widely used in linguistics to form analogous words such as grapheme, lexeme, tagmeme. Another linguistic ending that is phonetically similar is semic, derived from the word seme, and found in the words semic and polysemic. Keep in mind that there are other unrelated words that end in -emic with a short e, such as alchemic and pandemic.

Always tense:
-phobic (and the stand-alone adjective phobic)
-emic (phonemic, morphemic, etc.)
-semic (and the stand-alone adjective semic)

I also found two word endings that are "consistently inconsistent" in that all of the words with them have two alternative pronunciations, one with a long and one with a short vowel:

Sometimes tense:
-tropic (e.g. neuro-, ortho-, pleio-, pan-, photo-, psychotropic...)
-trophic (e.g. abio-, acido-, allo-, auxotrophic...)

Note that the stand-alone adjective tropic(al) can only be pronounced with a short vowel.

There are also quite a number of less predictable exceptions. It's hard to state generalizations, but I'll try anyway, first listing the words that can be put into some kind of categories.

Adjectives related to or derived from nouns where the vowel is long (the related noun is listed in parentheses after the adjective):

Always tense:                 Sometimes tense:
basic    (base)               genic       (gene)
phasic   (phase)              scenic      (scene)
rhotic   (rho)                splenic     (spleen)
psychic  (psyche)             spheric(al) (also hemi-, atmospheric)
microbic (microbe)(rare)      cyclic      (cycle)
disomic (disome)              saprobic    (saprobe)
velic (velum)                 palindromic (palindrome)
comedic (comedian?)           ccathodic    (cathode)
                              cellulosic  (cellulose)
                              centromeric (centromere)
                              allelic     (allele)
                              hygienic    (hygiene)
                              aldehydic   (aldehyde)
                              enzymic     (enzyme)(rare)
                              blastemic   (blastema)
                              presbyopic  (presbyopia)
                              choragic    (choragus)
                              parhelic    (parhelion) 
                              protic      (proton) (vowel usually long)
                              helical     (helix)

Adjectives derived from element names with long vowels (often those ending in -ium). This does not apply if an adjective of this form existed already before the element was named (e.g. titanic and germanic only have short vowels, even when serving as the adjective equivalents to titanium and germanium):

Always tense:         Sometimes tense:
chromic               ceric
bromic                niobic
rhenic                selenic
rhodic                ruthenic
magnesic              sodic (OED lists lax, but seems to be almost always tense)
                      uranic
                      vanadic
                      palladic
                      baric
                      thoric
                      chloric

Adjectives related to or derived from proper nouns with long vowels:

Always tense:         Sometimes tense:
Arcadic               Aeolic/Eolic
cretic                Adonic/adonic
                      Ethiopic
                      Hellenic
                      Salic
                      Priapic/priapic
                      Sothic
                      Romic

Adjectives related to or derived from nouns that have variable vowel length:

Sometimes tense:
pedagogic (the same variation applies to the noun pedagogue)
demagogic (the same variation applies to the noun demagogue)
phenolic (the same variation applies to the noun phenol) 

There are not many words that end in -esic, but a large portion of them seem to have pronunciations with long vowels:

Always tense:         Sometimes tense:
analgesic             amnesic (rare; amnesiac is more usual. OED lists a lax variant)
(coexists with        geodesic
lax-voweled           mesic  
analgetic)  

Miscellaneous words and word endings that are always pronounced with long vowels:

Always tense
aerobic
strategic
paraplegic (also quadri-, hemiplegic)
photic                              
lotic
thymic (also cyclo-, hypo-, euthymic)
sphenic
-chromic (achromic, monochromic, etc.)
hydric
nitric (interestingly, this vowel is also irregularly long in "nitrify")

Miscellaneous words where a long vowel seems to be usual, but a short vowel is listed in some dictionaries:

gnomic (seems to generally be tense) 
racemic (seems it is usually tense)
acetic (OED lists a lax variant, but it seems it is usually tense)
phenic (OED lists a lax variant, but it seems it is usually tense)
alethic (Oxford Dictionaries lists a lax variant)

Miscellaneous words and word endings that show variation and can have long or short vowels:

panegyric   

hypnagogic/hypnogogic
oxytocic
otic (also periotic; the vowel in parotic is usually lax, but can also be tense)
illeocolic
telic

xeric
steric (also allosteric)
metameric (related to metamer)
etheric (related to ether)   
systemic (related to system)
epistemic

neotenic
phrenic (also sometimes tense in hebephrenic, but generally not in schizophrenic)
(e)irenic(al)

clonic
nomic

Noachic

The Oxford English Dictionary has an interesting note for irenic:

In this and irenical adj., the first pronunciation [with a lax vowel] is that given by Smart, Ogilvie, and Cassell, and by Webster and the other American Dictionaries, and is in accordance with the general analogies of the language, as in academic, clinical, energetic, euphonic, Platonic, in which the long vowel of the Greek is uniformly shortened; but the modern use of the Greek Εἰρηνικόν, Eirēnicon, to which scholars naturally give the English academic pronunciation of Greek, affects the derivatives also, and makes the second pronunciation [with a tense vowel] frequent among university men.

Miscellaneous words where a short vowel seems to be usual, but a long vowel is listed in some dictionaries:

phosphatic (seems it is usually lax)
transuranic (MW lists a tense variant, but lax seems more common)

polygenic (seems it is usually lax)
photogenic (seems it is usually lax)
telegenic (MW lists a tense variant, but lax seems more common)

neurasthenic (MW lists a tense variant, but lax seems more common)
alkalotic (Collins lists a tense variant, but lax seems more common)
paretic (Dictionary.com lists a tense variant, but lax seems more common)

hyperaphic (Dictionary.com lists a tense variant; not in other dictionaries)

A note about my sources and abbreviations
Since I discuss the pronunciation of many words in this answer, some of which are are rare or have multiple pronunciations, I consulted various sources. My main sources for pronunciation are the following free online dictionaries: Oxford Dictionaries, Merriam-Webster dictionary (abbreviated MW), Collins English Dictionary, American Heritage Dictionary of the English Language (abbreviated AHD), Cambridge English Dictionary, Dictionary.com, and The Free Dictionary by Farlex; I'm accessing all of the above through OneLook Dictionary Search, which has useful options for wildcard search. I also looked at entries in the online Oxford English Dictionary (abbreviated OED), which requires a subscription, either individual or through an institution.