Wheel hop?
On my new to me 99 Bonneville with 82K miles I noticed when I'm making a hard left or right or even in a straight line takeoff (not flooring it but giving it more than the normal gas) it will chirp the tires then start to bounce/vibrate till I let off the gas, .(its embarrasing) I don't think this normal and if it's actually some sort of wheel hop or not... But I still need to know what to check or replace to get it to stop.

Check the lower control arm bushings, they are notorios for wearing out.

Good to know but what do they look like? I'm not very mecanically inclined. I could probably do it if I know what I'm looking for/at