A GTX 1060 or a RX 580 will run most modern games at decent settings with no problems, if you need a little cheaper you might want to check out older ones like GTX 980/970 or RX 480.
Edit: you could also buy a SSD for your machine, you'll really notice the difference!
It seems I'm looking at paying $200ish for one of those models. Does that seem reasonable? Also will my 450wat Psu be able to handle it or will that need to be upgraded too?
Yeah around 200ish is a good price. The PSU will do fine if it's a decent one. But, as the others said you should try replacing your HDD with a SSD first, you'll notice the greatest difference there, and if you still feel that your setup is still too slow for your needs you can start looking to replace the more expensive parts
I was wondering if my video card was at fault. Any recommendations for a budget card that will play games that are a few years old with little difficulty?