According to Jisho, かっ飛ばす means something like sending a ball flying among other things.
My issue is interpreting the verb when it directly affects people, like the following example:
私と君とでかっ飛ばそう
The only translated example I've seen of this verb is describing someone essentially knocking a ball out of the park.
かっ飛ばす is basically a baseball/golf term, but it can also work as one of those abstract verbs that mean something along the lines of "to go all out", "to smash", "to go full blast", "to rock it", and so on. For example, a rock musician might say かっ飛ばすぜ at the beginning of a live performance, in which case it has little to do with physically hitting a ball. Perhaps they are about to do something big or aggressive, but what they are actually doing depends on the context.
私と君とで just means "together with you and me". This で is the same で as in 1人で (alone), 全員で (with everyone), etc.
