アクセル全開で行こうぜ!
or
フルスロットルで行こうぜ!
or
アクセルペダル目一杯踏もうぜ!
Let's go at full throttle!/Full Blast!
should work.
I guess アクセルをべた踏みする is probably used when you need to accelerate very hard like slope. So, you might be told like that by the teacher at driving school.