Right now you seem to be using the integrated graphics on the motherboard, which is the ATI Mobility RADEON HD 3200, but you do have a pci-e 16x slot for a discrete video card. That is the good news.
There are lots of choices in video cards these days. You can look at this video card buyers guide over at Anandtech…its a few months old, but it will tell you what is available. http://www.anandtech.com/guides/showdoc.aspx?i=3538
Prices have slipped downward for many of those cards, and if you have $100 to spend on a card alone, then I’d recommend the Radeon HD 4850. Look for them at Newegg.com and Zipzoomfly.com. Be sure to measure the space in your case at the pci-e 16x slot and make sure you have room to fit such a card. The mid to high performance cards are pretty long.
The other thing to worry about is the psu. They were right in thinking your psu might not be able to handle a high performance card. The power supplies that are put into OEM machines generally are good for the equipment that they ship out with, and not much more. If you don’t want to invest in a new psu, you might want to look at cards that don’t need external power, like this HD 4670. Here is an example: http://www.newegg.com/Product/Product.aspx?Item=N82E16814161273
That card is one of the better ones I could find that doesn’t require the pci-e 6 pin power card.
If you do want to invest in a better card plus a power supply, look at Seasonic, Corsair, Fortron Source, and PC Power and Cooling as some of the better power supplies. 400-450w should be plenty since you are only going to run one video card in this computer. I run a HD 4850 in my machine and use a 400w Seasonic psu.
This Corsair psu is a great buy: http://www.newegg.com/Product/Product.aspx?Item=N82E16817139008
Edit: Check the make and power rating on your current psu before anything else though. Brokenbuga may be right in thinking it is adequate. I build my own computers and tend to over engineer them a bit in the power supply dept.