Here's my problem. When I step on the gas my car veers to the right, and then when I let off it veers to the left. Someone told me that it could be engine mounts so I changed those and it still does it. Has anyone had this problem before. I drive an 04 sunfire manual tranny.

This sounds more like a alignment issue.I to own a manual and if the alignment is out it would in fact pull to one side or the other.A simple check is to drive in a straight line on a level road at norm speed like 35 and apply ur brakes normally and do not hold the steering whl and see if the car drifts left/right.This is a oldschool technique to see if u may just need a simple alignment.If it's been more than 6 months since ur last align I will recom getting this done not to mention having ur whls balanced helps and torqued properly to spec makes a big diff.Granted if none of this is the case could be suspension like maybe new struts are needed.To check them push down on both sides of the front of the car if it bounces more than once ur struts r worn replace them this helps ur car alot espec in braking and body roll in curves.I hope this helps and if I left something off or incorrect may others add there knowlegde or correction.ok