The majority of the words that you have written in bold are a type of adverbial (a type of adverb) which is called an disjunct. (whether they are all disjuncts could be argued.).
.Disjuncts allow the speaker/writer to influence the hearer or reader.. They represent the speaker's attitude about what he is going to say.
Here are some examples where the adjunct is written in bold:
Naturally, you are going to go you. = the speaker is certain you will go.
Obviously,I agree with the president. = Its obvious that the speaker will agree.
Of course, she is late. = I think it is to be expected.
Regrettably, he didn't attend the meeting.= I think its sad or too bad that he didn't come.
Surely, you are going to go to school today. = a strong sense of persuasion.
Frankly, I couldn't care less about comma rules. = the speaker really doesn't care.
In each case we use a comma to separate the emotion or attitude of the speaker from the rest of the sentence. Some authorities believe that a comma should be used if the disjunct doesn't flow with the rest of the sentence and in most cases disjuncts are separated by a comma.
There are no hard and fast rules about comma use. and convention plays a role in determining some rules. You learn from experience and even English teachers can get it wrong.
Source Grammar 33 manual U of Saskatchewan, Longman's Dictionary, Guide to Grammar Writing by Charles Darling.
You do need a comma here. There is a systematic answer for this.
Whenever a sentence begins with an adverbial phrase, use a comma.
(source: https://owl.purdue.edu/owl/general_writing/punctuation/commas/extended_rules_for_commas.html)
An adverbial phrase explains things like: Manner, timing, reason, purpose among others. In this case, you have an infinitive of purpose as an adverbial phrase.
In your case:
To further our understanding of this phenomenon, (Adverbial of purpose)
we conducted yet another experiment. (main clause)
Other similar examples:
To help him start his car, we pushed it until it started rolling.
To better understand what they were doing, we started spying on them.
Usage:
These phrases use infinitives of purposes as adverbial phrases. It is much more common to place them at the end of a sentence like so:
We pushed his car until it started rolling to help him start it.
One might be tempted to think that it makes no difference, but it does. The reason one would put this at the beginning is to emphasize the purpose. Consider this context.
To help him start his car, we pushed it until it started rolling. To help him repair it, we called a mechanic. To convince him to leave, we offered him some of the stale food we had.
Also consider this phrasing in which the first purpose is logical, whereas the second oen is somewhat surprising:
We started spying on them to better understand what they were doing. To join them, we confessed that had been spying on them.
Best Answer
I would omit the comma unless you were using a quotation:
Yes, sometimes an introductory word will be followed by a comma; however, I don't think it's warranted in your example.