If you are trying for formal English, insert the implied 'to' in both your example statments:
I often go to watch movies with my friends.
I often go to watch movies with some friends of mine.
If you then omit the word 'watch' from these sentences they will become:
I often go to movies with my friends.
I often go to movies with some friends of mine.
These last versions of the sentences have a slightly different meaning, since they don't imply that you actually watched the movie, just that you were there.
If you are less concerned about formal speaking, your original example sentences work just fine.
The difference in meaning between your two example sentences is that the first implies all (or most) of your friends are present, while the second says that only 'some' of your friends are usually at the movies with you.
A listener might ask you who your friends are after either sentence, but they are more likely to ask you which subset of your friends you go to movies with in the case of the second sentence.