How to say “I (verb) to (verb)” in Japanese?

To make a verb into a noun phrase (which is what you want) you add の or こと after it. You then just treat that phrase as a noun, and use the appropriate particle after it, depending on what you want to say.
Formal and informal is no different, as you never end a sentence on it.

So in you case, 私は木曜日にNYCに行くことにワクワクする
However, that sentence is a bit unnatural sounding. ワクワクする tends not to be used like this, but rather stands as a separate phrase, for emphasis. 私は木曜日にNYCに行くから, ワクワクする would be better in this case. Or even just two separate sentences. 私は木曜日にNYCに行くんだ。ワクワクする。
And although in English when you say "I'm doing this on Thursday" it is assumed that you mean this Thursday, in Japanese it is more usual to actually state which Thursday. So "this Thursday" would be 今度の木曜日.
So 私は今度の木曜日にNYCに行くから, ワクワクする
Or instead you could say 今度の木曜日にNYCに行くのが楽しみだ (in this sentence, 私は is superfluous and therefore dropped).

As for saying "I like to do such and such", you use the same grammar I outlined at the start. So "I like to watch movies" would be 映画を観るのが好きだ

For polite forms, just change the sentence ending accordingly (だ for です, etc.)

1. Sadly, it's not that simple. It's a very different language.

2. For the first phrase, I suggest rewording to say: Going to NYC on Thursday is fun.

木曜日にニューヨークに行くのが楽しみです

(Thursday on New York City to go [act - which turns "go" into a noun --going] [subject marker] fun/pleasure/excitement is/will be.

Mokuyōbi ni nyūyōku ni iku no ga tanoshimidesu

tanoshimi desu -- is an idiom that expresses the idea of "I'm looking forward to" without using those actual words. (change desu to da if you want the plain form, like you did with iku in the original version)

Note that "I" is not actually expressed.

3. For the second phrase, note that "like" is not actually a verb in Japanese, but instead it's an adjective (better translated as "fond".

I am fond of watching movies = (as for me -- which is usually left out since you normally talk about your own likes) movies ]direct object marker] see [act - to turn see into a noun] [object of an adjective marker, which is also "ga", not just a subject marker. It's often called the exhaustive marker instead in both cases] fond am.

(私は)映画を見るのが好きです。

(watashi wa) eiga o miru no ga suki desu. (change desu to da if you want the plain form)

Note that there are many ways to express "I". watashi is the first one foreigners usually learn, but it isn't always the best translation.

3. 1 & 2 happen to be similar in structure, but that's not always the case. For example: (I) forgot to go = ikiwasuremashita (or - ikiwasureta in plain form).

いき忘れました . iki = connecting form of iku; wasureru - forget (dictionary form). The two verb roots directly connect to form a new verb (sort of).