Learn Russian: Давай завтра сходим в кино.

When saying "let's do something" with давай, the verb form you use depends on the aspect. For perfective verbs (completed actions), you must use the 1st-person plural ("we") future form, which is сходим here. You only use the infinitive after давай with imperfective verbs (for example, давай смотреть кино - let's watch a movie).

Why did the speaker choose сходим instead of пойдём to mean "let's go"?

Сходить specifically means to make a round trip—going somewhere, spending some time there doing an activity, and coming back. This makes it the most natural verb to use for attending events, going to the movies, or visiting a restaurant. Пойдём is also perfectly fine to use, but it focuses slightly more on the act of setting off.

Could I use давайте instead of давай in this sentence?

Yes, if you are talking to multiple people or to someone you address formally as вы. Because this sentence uses давай, we know the speaker is suggesting this to just one person they are on informal ты terms with.

Shouldn't кино change its ending because it comes after в indicating direction?