Using the polite form, I expect it to be something like:
Vielen Dank für lehren mich dieses Wort.
but I can't find any example thereof. What I have seen is "Vater, lehre mich..." translated to "Father, teach me to..." which puzzles me. Who is the subject here, Father, or me?
First:
German does not have those lean construction like for + verb-ing or Verb-ing, I verbed something else. You have to make normal subordinate phrases for most of those. The options here would be:
Ich danke Ihnen dafür, dass Sie mich dieses Wort gelehrt haben.
I thank you for that, that you taught me this word. (lit.)
Unless the word can cure cancer this is definitely too polite though :). The toned down version:
Danke, dass Sie mich dieses Wort gelehrt haben.
lehren is also very polite and formal in this example, so to tone it down to every day German:
Danke, dass Sie mir das Wort beigebracht/erklärt haben.
Second:
Vater, lehre mich...!
This is an order, thus it doesn't really have/need a grammatical subject in the sentence. Vater is just a pretext to address the order, it doesn't belong to the sentence and it is just there to let us know who is being talked to. Mich is accusative case of I.
In the context of your example, when you learn something with the help of someone you would use "beibringen" in modern German:
Vielen Dank für das Wort, das Sie mir beigebracht haben. (...das Du mir beigebracht hast) less formal
"Lehren" in this context would sound stilted and old-fashioned. As "lehren" would be an appropriate translation for "to teach" it is nevertheless used more in the context of a higher ranked teacher or even a university professor, e.g.
Prof. Schwäble lehrt Germanistik in Stuttgart.
Your example "Vater, lehre mich..." (likely followed by an accusative object) is probably from the context of the Biblical Father.
As a side note: in many regional dialects there is not such a clear cut difference between "lernen" and "lehren". The latter is sometimes not only used for teaching but also for learning.
I'd just like to add two even more toned down options for what you are trying to say:
Danke für die Erklärung!
or
Danke für die Info(rmation)!
Even if you've been taught exactly one word, those would still be totally appropriate. Referring specifically to dieses/das Wort will always sound a bit stilted.
When I first read the question I understood that it wasn't necessarily about "explicitly teaching (or learning) a new word", but perhaps rather about a situation that is actually quite frequent when learning a foreign language: You are talking to a native speaker (or at least someone who knows better than you), and you suddenly understand the meaning of a word that you hadn't understood before. In these cases you may say, for example:
Die "Gummistiefel" sind also eine besondere Art von Schuhen für Regenwetter?
...and when the person you're talking to confirms this, you could say:
Vielen Dank, du hast mir gerade ein neues Wort beigebracht.
...or even just:
Vielen Dank, ich habe gerade ein neues Wort gelernt.
If you want to point out, that you are glad to know this special word now:
Danke für dieses Wort.
But it is better to say thank you to the special person who told you that word:
Danke, dass Sie mir das beigebracht haben.
or, if you use "Du":
Danke, dass du mir das beigebracht hast.
If both, word and person is important, replace "das" by "dieses Wort":
Danke, dass Sie mir dieses Wort beigebracht haben.
The German word "lehren" (English: "to teach") is not very often used in German, and so even many native German speakers do not know how to use this word correctly. I try to give some examples for correct usage of this word in German (I am German native speaker, so maybe the English parts might contain errors, but German translation is correct:
The teacher teaches mathematics
If the teacher is male:
Der Lehrer lehrt Mathematik.
Better:
Der Lehrer unterrichtet Mathematik.
Female teacher:
Die Lehrerin unterrichtet Mathematik.
Next example:
My father taught me to swim.
Verbatim translation (which is correct German, but not used in real life):
Mein Vater lehrte mich zu schwimmen.
Better:
Mein Vater brachte mir das Schwimmen bei.
Here the verb is "beibringen", which is splitted into "bringen" (past tense: "brachte") and "bei". "Beibringen" has the same meaning as "lehren" or "unterrichten".
