H (t) = 16t^2+604 find how long it'll take to get the 348 feet

I believe you meant - 16t^2 + 604 ft. If you leave that - sign out, this function will never equal 348 feet.

-16t^2 + 604 = 348 can be solved:

-16t^2 = - 256 ft, and t = 16 sec. Whatever you are throwing will first rise, reach some max. value, and then begin falling, and eventually reach 348 ft.