The key is whether "you and I/me" is the subject or the object of the verb. Here, we are going to "grab a bite", so "you and I" is appropriate. We could rewrite the sentence as:
Let's you and I grab a bite to eat.
On the other hand, if "we" are the object of the sentence, use "you and me".
She gave the money to you and me (to buy a house).
This cake was made for you and me.
Could you go get some lunch for you and me?
However, before you get too excited, and think you understand it completely ... unfortunately many English speakers don't know or don't care about this grammar, and instead use whatever sounds best:
Let's go for a walk, just you and me alone.
I think, you and me, we can build it together.
It's true we'll make a better day just you and me
Related article at ELU