I built my first computer about 2.5 years ago with the help of a friend who is way more tech savvy than I am. I'm getting to the point where I think it's time for a bit of a upgrade, as I'm seeing frame rates dropping off noticeably in newer games coming out. Here's what I'm currently using:
GIGABYTE GA-Z68AP-D3(R2.0) LGA 1155 Intel Z68 HDMI SATA 6Gb/s USB 3.0 ATX Intel Motherboard
CORSAIR Enthusiast Series TX650 650W ATX12V/EPS12V 80
EVGA SuperClocked 01G-P3-1563-AR GeForce GTX 560 Ti (Fermi) 1GB 256-bit GDDR5 PCI Express 2.0 x16 HDCP Ready SLI Support Video Card
Seagate Barracuda ST1000DM003 1TB 7200 RPM 64MB Cache SATA 6.0Gb/s 3.5" Internal Hard Drive Bare Drive
2 x Patriot Gamer 2 Series 8GB (2 x 4GB) 240-Pin DDR3 SDRAM DDR3 1333 (PC3 10666) Desktop Memory Model PGD38G1333ELK
The graphics card was one of the pieces I choose not to spend as much on at the time, but it was a solid card nonetheless, just starting to show its age I think.
I was looking to spend no more than $500 dollars. I think I may also need to update my power supply, as it seems like mine doesn't appear to meet the requirements for some cards out there today, so that's a cost I'd have to factor in (and wouldn't mind suggestions there too). The GTX770 seemed like a well reviewed card, but I'm not too good at comparing graphics cards to be honest, so I could use the help. Ultimately, if the value is there, I wouldn't mind going outside my budget some.
You're correct with the GTX 770, and it is highly recommended.
I've gathered together a few cards that are in your budget.
The GTX 770 and R9 280X are at equals, and so are the 780 and R9 290.
The GTX 770 and R9 280X are actually the same price, and are substantially cheaper than your given budget.
If you can do an extra $100, you can buy two and run them in SLI. That will then be a VERY strong graphics solution.
If you would prefer a less-strong, but still in your budget single card solution, then the R9 290X takes the cake. It competes with the NVIDIA GTX 780TI, which is much more expensive than your budget, and is only slightly out of your budget.