After the game, we got something to eat. Are the phrases after the game and to eat? And if yes, are they both adverb phrases?

Answer 1

to eat = an infinitive (a verb form)

After the game - yes, it's adverbial.

Answer 2

so to eat is not an adjective or an adverb phrase right?

Answer 3

right

Infinitives can be parts of phrases, but they themselves are not phrases.

http://grammar.ccc.commnet.edu/grammar/phrases.htm#infinitive