My car sputters when I'm sitting idle with it running? What's wrong with it?

My car sputters when I'm sitting idle with it running and sometimes when I try to accelerate from a slower speed. I have a 2003 Kia Spectra. If sitting in idle at a light, it seems to calm down if I put it in neutral while I wait. It also doesn't seem as bad when the fuel tank is more full than when it's getting low. It's not always too bad, but it seems to be getting more frequent lately. It also shakes a lot on the highway, almost like my seat and the steering wheel and the whole front end are vibrating really hard (in case that might have anything to do with it). We just changed the fuel pump last weekend and going to change the fuel filter this weekend, but we had hoped the sputtering was caused by the bad pump. What could be wrong?