I noticed when I'm making a hard left that the wheel seems to vibrate violently. Is this normal?

Its done it twice now, once when I was making a U-turn and got on the gas a little quick. And then again today when I was turning onto a side street but pulled up a little to far and had to make a sharp turn.

Not normal. Sounds like a CV joint going south.

No not a CV I don't think. There is no clicking or clunking just shaking.

I noticed today that if I floor it, it gets going but when it changes to the next gear it shakes too. I'm thinking it'* either the shocks/struts or the motor/tranny mounts..

Anyone else know what it could be?

Ok it seems to doing this a little more often now(left and right) It will chirp the tires then start to bounce/vibrate till I let off the gas, Even in a straight line takeoff.(its embarrasing)

Maybe cause I'm pushing it a liitle more now that I'm a little more used to driving it?

But I still need to know what to check or replace to get it to stop.

Not being a smartass but are your lug nuts all tight? If they are then push down on the front end of your car on each corner if it moves a lot then your struts are bad. You can also check your endlinks because mind sounded the same way and the bushings were gone on the endlinks, they are only about $11.00 from Autozone. Hope this helps.