Upgrading guidance (gaming) 1366x768 display, 300-watt supply

I should notify you before we begin that I am looking for the cheapest possible solution.

I recently acquired a Dell Inspiron 660 with:

Intel Core i3-2120 3.30Ghz

6 gigs of RAM

Windows 7 home premium (64-bit)

Integrated Intel HD graphics

300-watt power supply

Obviously I didn't buy this machine expecting to have a top-of-the-line gaming rig out of the box, but I am planning to transform it over time into a decent and perhaps even high-end rig (if practical/possible). I'm not a computer expert, so bear with me if I start to sound like I don't know what I'm talking about.

Anyway, what I want to focus on in this discussion is the graphics (NOTE: I'm using ONE monitor with a 19", 720p screen, with a max resolution of 1366x768, so let's work within those parameters). I realize that my power supply is limiting, but I know it can be upgraded, if I'm not mistaken. However, I'd prefer to keep it as simple as possible. Is there a graphics card in existence that could run most new games (e.g. The Witcher 2, Skyrim, Arkham City) on medium or possibly high settings at a good frame-rate with my power supply? If so, that would be the way to go. But like I said, I am willing to upgrade if necessary.

So, considering all of that, what are your recommendations for graphics cards?

Any help and advice would be greatly appreciated, and feel free to convey any additional advice or wisdom if it will help my cause.

Thanks in advance,
Alyosha.

P.S. Including prices in your replies would be nice.