Replace the head unit first, that alone will improve sound.Get a good 2 channel or monoblock to drive a single sub and that should be enough for what your looking for.If you want more replace the stock 5X7's with components in the fron and 2 ways in the rear and add a clean 4 ch. amp to power the front and rear speakers.Probably going to end up going over your budget though, most of it from the head unit.

 

I just installed a 4ch 300W Alpine amp and a 10" Alpine sub in the trunk for around $200. Had to rewire most of the car because of the Shaker HU but the sound is incredible! I will be adding components in the front doors that are 2ohm because with the rear deck speakers connected (which are 2ohm and fronts are 4ohm) they are stealing all the power and sound. I like my highs in the front and bass/fill in the back. Either way, the stockers have good sound when you give them power.

 

I agree, i put a 85X4(160X4 maxx) on the stocks and they sounded as good as the components in put it.