2000 Pontiac Grand Am A kind of sputter?

My girls' car, the one listed, is almost sputtering when trying to accelerate up a hill. It will get just above 2k rpm and will start jumping back to about 1800 rpm and 2200 rpm. It is a little strange to me. She insists ever one she has talked to says it is transmission, but I have a feeling it is fuel. I bought a fuel filter to install, but haven't gotten around to it yet. Also, when you push a little harder on the gas, once it starts doing this, it usually stops happening, but shifts a bit harder. And the problem only happens going up hill. I appreciate the help!