The most important thing is to get a good stand-alone video card. Do not get anything that doesn't have dedicated video graphics. Even the newer adventure games are requiring dedicated video cards.
I tried to put something together at Dell, but didn't have much luck in specifying a video card. I didn't spend too much time there though. The other thing is to be sure a good enough power supply is included - most of the low-end desktop computer from manufacturers like HP and Dell have integrated video and puny power supplies. But you can look at the different manufacturer's websites to see what they offer.
I'm going to post component specs for a decent computer that will still work for you from EcollegePC. They will build one for you for about as much as you could do it yourself. But you con't have to order from them... this is just an example. You might also look at Cyberpower - they also build PCs and sometimes have some pretty good deals.
I'm assuming you want Windows 7, not Windows 8 and I only put in 4 GB of RAM and a 500 GB hard drive. You can always add more RAM and a bigger hard drive if you want. The motherboard isn't "cutting edge," but is fine for your type of usage - something else you can upgrade if you want. I'm also assuming you didn't need a new monitor, mouse or keyboard. This configuration is $761.
Intel Core i5-3450 3.1Ghz 6MB Cache Quad-Core Stock Intel LGA1155 Heatsink and Fan
GIGABYTE GA-H61M-S2H (Intel H61, HDMI, DVI, PCI-E, 4xSATA, 2xDDR3)
4GB (2GBx2) PC3 10666 DDR3 1333Mhz Memory Lifetime Warranty
500GB 7200RPM 32MB Cache SATA3 6Gbps (Major Brand)
24X LG SATA Dual Layer DVD /-RW/CDRW w/Nero
1GB ATI Radeon HD7750 GDDR5 PCI-E DVI/HDMI (Major Brand)
Microsoft Windows 7 Home Premium 64bit
Raidmax Blue Super Huricane (3 5.25, 4 3.5) 2 Fans, Audio/USB 3.0
500watt Cooler Master Extreme Power Plus | Onboard LAN included
Onboard Sound included
Wires and Cables neatly tied up away from fans
3 Year Parts and Labor Warranty
EDIT: Put this together at HP for $829. It's the only one I could do that still had Windows 7. At least HP allowed a power supply upgrade which I was unable to find an option for at Dell.
Operating system Windows 7 Home Premium 64
Processor 3rd Generation Intel(R) Core(TM) i7-3770 quad-core processor [3.4GHz, 8MB Shared Cache]
Memory 8GB DDR3-1333MHz SDRAM [2 DIMMs]
Hard drive 1TB 7200 rpm SATA hard drive
Graphics card 1GB DDR3 AMD Radeon HD 7570 [HDMI, DVI, VGA via adapter]
Power Supply 460W Power supply
Primary optical drive SuperMulti DVD Burner
Networking Wireless-N LAN card (1x1)
Productivity ports 15-in-1 memory card reader, 2 USB 2.0 (front), 2 USB 3.0 (top)
Office software Microsoft(R) Office Starter: reduced-functionality Word & Excel w/ ads. No PowerPoint or Outlook
Sound Card Beats Audio (tm) -- integrated studio quality sound
Keyboard and Mouse Premium HP keyboard and optical mouse
Included Software Bundle Adobe Premiere Elements & Photoshop Elements 10
In my PC corner for the winter.