The most obvious answer is, just keep doing heroics and deal with the elitist hate.
Other than that, Strength is your primary stat. Period! So your weapon (agility) and your relic (intellect) need to be replaced with items with Strength. In addition to that, your helmet and one of your rings are itemized for tanking (Dodge, Parry, Block).
You want to focus on replacing your poorly itemized pieces asap, as well as the crafted pvp items you have equipped (Gloves, Belt, Pants). The stats you want the most are: Strength (Over everything else), Hit (To Cap), and Expertise (To Cap). At this point Mastery, Crit, and Haste are next in line. If you want to read more on this check out the post below.
As far as talents go, your 3 points in Clarity of Purpose are completely wasted and would be much better spent in Seals of the Pure (Tier 1 of the Protection Tree). You shouldn't be healing yourself at all in a dungeon (And if you absolutely have to, you should be using a 3 point Word of Glory).
I would personally replace Glyph of Judgement with Glyph of Templar's Verdict as it will yield a much higher dps increase than Glyph of Judgement.
I would recommend that you try and find a guild with friendly members that will run dungeons with you while you learn your rotation and collect better gear.
If you are not an addon purest and would like a rotation helper, CLCRet is a great tool to help you learn your rotation. It keeps track of all of your cooldowns and buffs and tells you via an icon what ability should be used next to continue your optimal rotation. It isn't perfect, but it will help you learn timings (like making sure Inquisition doesn't fall off).
If you want to read some really good information about Retribution Paladins (Specs, Rotations, Glyphs, Stats, Reforging, Gemming, Enchanting, Cooldowns, etc) I recommend this post on Elitist Jerks.
The heirlooms are super effective items. They are equivalent to some of the best dungeon drops you can get(or better) and they keep up every time you level.
Spirit only matters in long fights (five minutes or more) such as raid bosses. It shouldn't matter for dungeon instance running. You should manage your in combat/out of combat time so you can drink if your tank is chain pulling. Sometimes it's worth it to allow the tank get out of range so you can finish your drink.
Resist the temptation to heal stupid. If the tank pulls before you're ready, he deserves swift death. If the dps pull before the tank is ready, they deserve swift death. You don't have to kill bad party members, but you don't have to save them either. If you try to save everyone from their mistakes, you will run out of mana.
I put together the efficiency statistics for my 68 holy priest that has a couple of heirlooms and no WoTLK gear (this research ignores crits, and considers chakra:serenity and Inner Fire). You might be able to use this information to choose more efficient heals and better keep up with the group.
Cast Time Heals
Heal 144 mana 2.45 cast 6792 heal 47.1 h/m 58 m/s 2772 h/s
Greater Heal 448 mana 2.45 cast 14504 heal 32.3 h/m 182 m/s 5920 h/s
Flash Heal 448 mana 1.47 cast 10878 heal 24.2 h/m 304 m/s 7400 h/s
Heal is super efficient and should be a go-to choice. Low heals per cast time is the downside.
Flash Heal is super inefficient and should be a last resort (except when it's free).
Instant Spells
PW Shield 464 mana 1.47 gcd 10781 absorb 23.2 a/m 315 m/s 7334 a/s
Renew 198 mana 0.97 gcd 1185 + 1974x4 heal 45.8 h/m 204 m/s 9361 h/s
HW Serenity 152 mana 1.47 gcd 8583 heal 56.4 h/m 103 m/s 5838 h/s
Power Word: Shield is super inefficient - last resort. Holy Word: Serenity is the go-to rescue spell. Renew should be kept up on tank, as it can be extended free by any direct heal via Chakra: Serenity.
Multitarget Heals
Binding Heal 410 mana, 1.47 cast 7880x2 heal 38.4 h/m 278 m/s
Prayer of Healing 342 mana, 2.45 cast 3861x5 heal 56.4 h/m 139 m/s
Circle of Healing 243 mana, 1.47 gcd 2687x5 heal 55.2 h/m 165 m/s
Prayer of Mending 266 mana, 1.47 gcd 3285x5 heal 61.7 h/m 180 m/s
These spells would be mana efficient if everyone took damage at the same rate as the tank. These are group rescue spells. Their efficiency is comparable, so use the one that fits your time and aoe shape (you+1 vs all). Slight nod to Prayer of Healing for low mana per second.
Other stats:
Int: 454, Mana: 7600
Spirit: 239, non-casting mana regen: 421 mana / 5 seconds.
Informally observed mana regen rate: ~170 mana per second when not casting.
Tanks at this level typically have ~12k hp.
Best Answer
You will get a lot of gear through the initial stages of questing that will increase your item level, this is the quickest 'cheap' way to achieve the required item level to enter these dungeons. The quickest 'overall' method of getting higher level gear would be through using the auction house, although other options include levelling the tailoring profession and using that to create cloth gear for questing/dungeons.
Depending on the gear that you had in Cataclysm, you should be able to enter the initial Temple of the Jade Serpent and Stormstout Brewery instances straight away but if you do not meet the requirement for these then the upgrades you get from the initial quest chain in Jade Forest should dramatically increase your item level quite quickly.
Average item level is calculated by taking the item level of each item you have equipped and dividing it by the number of equipment slots excluding the tabard and shirt slots. In the instance of using a two handed weapon, the off hand weapon slot is not included in the calculation. However, in the instance where you have a 'required' slot empty (ie: head slot or shoulder slot) this slot is still included in the calculation but with an item level of 0 when summing up the item level of that slot. This means that a fully equipped character using two one hand weapons has sixteen item slots while a character using a two hand weapon has fifteen item slots.