Windows 7 - Looking to upgrade my computer

I'm using the Gateway DX 4710 (Gateway Support - DX4710*Gateway DX4710 Desktop Series) as my main computer. I want to be able to play most modern games on it, but I'm not sure if it's worth upgrading this computer I got in 2008.

My system specs are (Let me know if I missed anything):
Intel Core2 Quad @ 2.40GHz
6 GB of DDR2 RAM
Intel G33 Express Chipset
Gateway G33M05G1 Motherboard

Of course, my main worry is getting a new video card.
As for games, I'd like to play Battlefield 3, Skyrim, and any other 'modern games' at low-medium settings at 1680x1050 resolution and an average of 30+ FPS.

Any recommendations on what I should upgrade on my computer?
My budget is around 300$, but I'd like to stay as low as possible.

Thanks.

That is a good computer. All you need is a good graphics card. And for up to $300 you will have no trouble finding a good one.

You could probably go with an MSI GTX 560 Ti Hawk. They're selling for around $265 (in America) and it is an extremely good card. As you can see in this high-end GPU list, it's ranked 9th and overshadows most other cards.That particular card also comes with a 2 year warranty.

However, if you want a 3 year warranty and slightly better performance you could go with an EVGA GTX 560 Ti. It's selling at just under $300 but it's a solid card that will be able to play most if not all games on high settings.

Interested in getting the MSI card, what recommended PSU should I get?

On Newegg they do not specify the power requirements. But it is safe to assume that a stock PSU will not do. Get a nice Antec 650W and you will be safe. Also make sure it fits into the space. The thing is pretty big.

I suggest CM Silent Pro M600. Silent Pro M600 - Cooler Master - Leading Provider of Computer Case | Cooler | Power Supply
Will be enough for that conf.

If you're wanting to stick two a card under $200, I would go with the SuperClocked version of that card.
Newegg.com - EVGA SuperClocked 01G-P3-1461-KR GeForce GTX 560 (Fermi) 1GB 256-bit GDDR5 PCI Express 2.0 x16 HDCP Ready SLI Support Video Card

If not, take a look at the two cards I recommended in my previous post.

You should also take a look at the below thread. It has a lot of useful tips for anyone planning to upgrade their GPU.
Planning to upgrade your Graphics Card? Read First...

For a GTX 560, you'll need at least a 500w PSU. I would go for an Antec 650w - 800w to be sure. Alternatively, you should look at the combo packages that sell PSU's and graphics cards together. In particular look at the following deal.
Cooler Master Silent Pro Gold Series 800w w/ MSI GTX 560 Ti Twin Frozr III

I guess I'm set on the EVGA SuperClocked GTX 560 and the CM Silent Pro 600W PSU.

Frankly, I'm not experienced with installing computer parts, so I'm not quite sure on how to put in the PSU and the graphics card.

Piece of cake. Last time I did it :