Kumusta! – Strictly speaking, “kumusta” should be followed by a question mark rather than an exclamation point because it means “How are you?”.

Ang pangalan ko ay Nate. – Tama (correct)

Nice to meet you! ->  “Ikinatutuwa kong makilala ka!” = It pleases me to meet you/I am happy to meet you! – “Ka” is “you” (singular, informal).  Use “kayo” instead if you’d like it to be formal or if you are addressing an older person.  “Kayo” is also what you use to mean “you” (plural).   

Gusto kong matuto ng Tagalog dahil mamamasyal ANG aking pamilya sa Pilipinas sa Disyembre. – “Mamamasyal” usually means “to take a leisurely walk”.  “Mamamasyal kami sa park” = We’d take a walk in the park.  It would be better if you’d use “papasyal” (to make a short visit) instead – “... dahil papasyal ANG aking pamilya...”.  You need the article “ang” to precede the common-noun subject (my family) of your sentence. 

Kahit na maliit lang, gusto kong makipag-usap sa aking mga kamag-anak.  -  Correct, except for “maliit” (small, little), which refers to size, not quantity.  “Kaunti” (few, little) is what you use for quantity.  “Kahit na kaunti lang” = Even just a few/little.

Ito ang magiging unang pagkakataon ko pupunta sa Pilipinas. – Correct, except that you will need a linker (na/ng/g) for “ko” to relate it to the following phrase; and “pupunta” (going to) should either be “pumunta” (to go) or “makapunta” (to be able to go). => Ito ang magiging unang pagkakataon kong (or “ko na”) pumunta/makapunta sa Pilipinas.

Nuno ipapakita ko sa inyo. – “(Mga) ninuno” is “ancestor(s)”.  I do not know if “mga nuno” is used to mean “grandparents” in some Tagalog regions, but here in Manila, we don’t use “nuno” to refer to them lest we may get disinherited.  :-)  “Nuno” is a word we associate with old supernatural creatures of folktales - think of Tolkien’s Middle Earth.  We’d say instead, “lolo at lola”, if we are referring to a single pair, and “mga lolo at lola”, if we mean more than one pair.  “Mga lolo at lola (ko), ipapakita ko sa inyo.” = (My) grandparents, I’ll show you/watch me.

Ako ay maraming matuwa! = I am pleased plenty!  -  I suppose what you meant was, “Many will be pleased with me”, in which case, that would be, “Maraming matutuwa sa akin.” Also possible, “Ako ay makapagpapatuwa sa marami.” = I will be able to please a lot (of people), but that is, if you can manage a 7-syllable word at this stage.  :-)

Hello!

Kumusta! Ang pangalan ko ay Nate. Nice to meet you! Gusto kong matuto ng Tagalog dahil mamamasyal <em>ang </em>aking pamilya sa Pilipinas sa <em>darating na</em> Disyembre. Kahit na <em>maigsi</em> lang, gusto kong makipag-usap sa aking mga kamag-anak. Ito ang magiging unang pagkakataon na makapupunta sa Pilipinas. Nuno ipapakita ko sa inyo(?). Ako ay <em>natutuwa!</em>