I was in the same predicament/budget as you and was given some good recommendations here. While I was recommended a tower, I wanted a HTPC case to fit along side my other home theater components in my TV stand. In the end, I wanted to be able to play Diablo III, Starcraft 2 and WoW on max settings. PC is hooked up to my 64" 1080p plasma (http://www.samsung.com/us/video/tvs/PN64D7000FFXZA) through my Denon AVR. Currently playing beta for Mists of Pandaria on ultra settings and it is awesome.
Initially, I was going for a prebuilt, but luckily others have shown me the light. It was my first build ever, but with help from here I grew the confidence to pull the trigger and do my first build. Another deciding factor was that the GTX 680 was finally in stock too hehe. Had a few minor snags, but all turned out well.
I bought all my parts from newegg.com (copied/pasted from my wishlist):
Intel Core i7-2700K Sandy Bridge 3.5GHz (3.9GHz Turbo) LGA 1155 95W
COOLER MASTER GeminII S524 120mm Long Life Sleeve CPU Cooler
ASUS GTX680-2GD5 GeForce GTX 680 2GB 256-bit GDDR5 PCI Express 3.0 x16
ASRock Z68 Extreme3 Gen3 LGA 1155 Intel Z68 Motherboard
SAMSUNG 830 Series MZ-7PC256B/WW 2.5\" 256GB SATA III MLC SSD
Crucial 16GB (2 x 8GB) SDRAM DDR3 1333 (PC3 10600)
XFX PRO850W XXX Edition Semi-Modular 80 Plus Silver Certified 850 Watt Active PFC Power Supply
LIAN LI Black Aluminum PC-C60B ATX Media Center / HTPC Case
Microsoft Windows 7 Home Premium SP1 64-bit
ASUS DRW-24B1ST/BLK/B/AS Black SATA 24X DVD Burner
People will tell you that an Intel Core i5 2500K (or 2550K) will be enough for today's games. I opted for an i7 2700K for semi-foolish reasons and in case I use the PC for other things down the road. Plus, I read an article here that MMO and RTS type games do benefit from a better CPU (or maybe it was in my dreams?).
As for the HTPC case itself, all of my components fit. People may not like the CPU cooler I've chosen in particular, but it's perfect for the HTPC case and CPU that I got. Everything is a less than snug fit in the case since I opted for an 850W power supply (I was recommended a 750w one). I took out the pair of 3" bays since I don't need them which also gives my case better airflow it seems.
For hard drives, I opted for my own choice of SSD based on reviews. It's fastened on top of the bay for my DVDr drive. 256GB is enough space for the OS and the amount of games I want to play.
Eventually, I do plan on replacing the pair of case fans to something quieter hopefully. Also, installing acoustic sound deadening material may be in the future, but the PC noise is more than bearable at the moment. I only hear it when I'm not playing a game and the PC is just idle. I don't overclock anything so I don't plan on adding a 3rd case fan (on top) for now. Everything runs under 100 degrees Fahrenheit.
Hope this helps. I can't really offer any recommendations as I still consider myself a newbie, but it was a great experience nonetheless.
Including tax and shipping, it came out to around $2100-2200. Cost went up due to me wanting a slightly bigger power supply, the SSD went up in price when I was ready to pull the trigger and I forgot to factor in the cost of buying Windows 7 OS.
Building a PC really isn't that intimidating after your first one. I was surprised at how simple it was. It was more so time intensive since I was being more careful.
Just do some searches for installation/building videos for PCs and components. It's also great to have a friend or coworker with you to help you with the process. Plus, you'll have forum members from here to help you out as well. Good luck!