Well right now your processor is a major bottleneck, and to change that out you'll need a new motherboard so basically a whole new machine. You really should move into the newest generation of intel i5's or if you do a lot of multi-threading applications like video editing or pictures an i7 would be a better choice. An i5 4670k would probably do pretty well and give you the option of doing some light overclocking later if you wanted. Your graphics card is also pretty old.
I would advise you to avoid buying a system from a company like Dell, HP, eMachines, etc... They make computers for mostly business use, they don't have the room for upgrades or aftermarket parts that something custom built does.
A decent little gaming system can be pretty cheap if you don't go crazy with it. Here's a link to a decent little gaming PC for about $800 which you could go higher or lower on depending on what you get. http://pcpartpicker.com/p/3aAaC
Something like that should be pretty good for at least 3 or 4 years though. Not going to be setting any records but definitely able to play any of the newest stuff on at least medium quality.