Agree with the above, for a cheap and cheerful upgrade a HD7750 / 7770 would be a good idea, I used the HD7750 in a very similar spec Advent that I recently got rid of. The HD7750 managed skyrim and Farcry 3 at playable levels. I guess anything more powerful and you might start to run into issues with bottle-necking etc.
You could get a second hand one on Ebay for £30 - £50. Cheap cheerful upgrade. Failing that build a new budget Gaming PC for around £400? All depends on your budget I guess!
A Phenom II X4 945 should be fast enough to keep up with most graphics cards imo.
Personally I think the best value for money cards about at the moment is the Radeon R9 270. Going much higher than that with that CPU wouldn't be worth it, and a 270 can run pretty much anything on max settings at 1080p.
Other things you'll need to consider- how much ram does your system have- you really should have at least 8gb these days. Also you need to make sure your power supply can handle the card, as unknownofprob said it's important to get the PSU correct (there are allot of cheap supplys that state high wattages however they just give up when you plug a fast graphics card into them, you need a decent branded supply with a single, high current 12V rail e.g. 1 x 12V @ 40A, rather than 'split rails' that the cheap supplies use e.g. 2 x 12V @ 20A).
Edit: based on your updated post- a Geforce GTX 750ti is probably your best bet. Its the faster bus powered card available. There is also the GTX 750 (non ti). Both cards are fast enough to run modern games well at 1920 x 1080 resolution at medium to high settings and are the most power efficient cards on the market right now. You may still need to look at putting a better PSU in there though.
the GPU (GTX 760 will have a huge impact on gaming performance and other tasks. You'll have to buy both, as the GTX 760 has a TDP of 170W, and your CPU has a TDP of 125W, already that's 295W (not including ram, HDD/s, motherboard/CPU cooler ect..), you won't be able to test it with a 300W PSU.
Your motherboard has a PCI Express x16 slot (best it can be), which will let your GPU go to it's maximum performance.
Your case needs to support this card, so make sure it has 270CM+ of length from rear of case to HDD bay.