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?

to eat = an infinitive (a verb form)

After the game - yes, it's adverbial.

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

right

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

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

Yes, the phrases "after the game" and "to eat" can both be considered adverb phrases.

To analyze these phrases, we can break them down into their constituent parts.

1. "After the game":
- "After" is a preposition indicating temporal relationship.
- "The game" is a noun phrase, where "the" is the definite article modifying the noun "game".
- When "after" is combined with "the game", it forms a prepositional phrase modifying the verb "got".
- The phrase "after the game" answers the question "when did they eat?" and functions as an adverbial phrase modifying the verb "got".

2. "To eat":
- "To" is a preposition signifying purpose or intent.
- "Eat" is an infinitive verb form, which functions as the object of the preposition "to".
- When "to" is combined with "eat", it forms an infinitive phrase indicating the purpose of the action.
- The phrase "to eat" answers the question "why did they get something?" and functions as an adverbial phrase modifying the verb "got".

Therefore, both phrases in this context can be identified as adverb phrases modifying the verb "got".