Yes, it is grammatically correct and very good English. Be aware, however, that 90% of American English speakers will say "Do you know who you'll go with" -- even though "who" is grammatically wrong. Some people will think that, when you say it correctly as in your sentence, that you sound uptight (overly formal).

It's OK but using the word "whom" might be thought of as pretentious. People don't normally use "whom" except when writing a formal document. I might have some language experts angry at me when I say this, but I would recommend when speaking informally just say something like "Do you know who you will go with?" ... Yes, it ends in a preposition but who cares? It will be easier to understand. On the other hand if you're going to use this phrase in a paper you will turn in for a grade, say something like you said. But on the third hand ... I dislike "whom" so much that I would avoid it even in a formal setting and say something like "Who is going with you?" This doesn't end in a preposition and doesn't sound pretentious.