For a console gamer that's making the jump over the PC, the thought of building your own PC is a bit too daunting (although it's the most cost-effective option)... However, the release of Steam Machines will likely be the more desirable option for your transition.
Steam Machines are console-like PC's that are primarily designed for gaming. They're usually controlled with just a game controller, but can be run with a keyboard + mouse / trackpad. Right now there's two major manufacturers of Steam Machines...
* Alienware Alpha -- http://www.dell.com/us/p/alienware-alpha/pd
* Syber Steam Machine -- http://www.sybergaming.com/products/steam-machine.aspx
While the specs between the two are fairly similar, I've personally went with an Alienware Alpha (i7 configuration) as I knew the brand (& I was able to get a discount from Dell through my employer. ^_^). It's actually the system that was used to write this answer.
The Alienware Alpha originally came in 4 configurations (i3 base, i3 enhanced, i5 & i7), but for some reason they removed one of the configurations for some unknown reason. The different configurations have different specs (processor, system RAM, HDD space) & cost, but all of them include a 2 GB custom graphics card. The Alpha runs on Windows 8.1, but is upgrade-ready to 10, & includes it's own custom software to use it like a console. It comes with a wireless XBox 360 controller WITH wireless USB adapter. The system is a little larger than a Nintendo & the hardware sets up very similar to a console (15 minutes or less, depending on how much work you need to do moving stuff around your media center). It connects your TV via HDMI port (which the system does include a pass-through port, so you don't what to worry about not having an open port for the system) & can be operated with the included controller. HOWEVER, I highly recommend buying a wireless USB keyboard + trackpad / mouse combo like the Logitech K400r (which runs ~$30 at Walmart), which you can hide the connector within the system itself (as there's a hidden USB port on the underside of the system, accessible behind the removable panel).
As for the custom software, it was originally bare-bones as the "Alien UI", but they've overhauled with & released the "HiveMind UI" as it's replacement, which add more functions (like music & video playback & streaming support). However, you can always jump into Desktop mode once you have a keyboard + Mouse / Trackpad connected to it. Beyond the UI software & Steam, the system is pretty bare-bones in terms of software as it only includes some maintenance stuff (like Alienware Respawn, so you can restore your Alpha to factory settings, should you mess things up) & NO practically no bloatware... So if you do tread into Desktop Mode, pick up an anti-virus program (like AVG or Avast... which are both free for personal use) to keep you safe as Console Mode is more of a walled garden (you can only access items that are made for the Hivemind software & Steam).
The Alpha supports 720p & 1080p resolutions, which I have mine set to 1080p as my HDTV supports it... but you can run with 720p as well if you like. As for performance, this will vary based on the configuration you go with, but my Alpha i7 pretty much runs all the games that I've thrown at it at default settings with a framerate of at least 30 fps or better. Minecraft generally runs at 60 fps (although the throttle is set to 120 fps) & GTA V with higher-than default settings run around 30 fps (as it's the only graphically intensive game I've actually bother to do any benchmarking with) @ 1080p. Framerates would like improve if I stuck with 720p. The only downside that I have to mention is that the GPU in the Alpha is non-upgradable, so you may start running into some performance issues with newer games in about 5 years (lowering the graphical settings in those games will likely help, but won't look at great)... but as with consoles, there'll be a new generation out by that time with higher specs, should you wish to stay on the upper-end of PC gaming. Personally, I'm planning to get about 10 years (potentially more) out of my Alpha before it officially gets permanently retired (it may get semi-retired after 5+ years, depending on my gaming needs).
As for games, most of them will work with minimal issues while in console mode. HOWEVER, a few older titles (which require extra downloads of legacy files) may not work until you address the issues in desktop mode (GTA, GTA2 & Bionic Commando: Rearmed are the only 3 titles in my Steam Library that forced me to visit desktop mode to get working so far... all others have worked fine without issues).
While the i7 configuration of the Alienware Alpha is on the high end of your budget (retailing at ~$900, although you way want to shop around for a better deal OR check with your employer if they have any discounts available through Dell to help lower the cost), it's a great system & would definitely make your transition from console to PC a bit easier.
I know this may sound a bit like an ad for the Alienware Alpha, but I'm happy with the system... even though I could have build my own for less (but I was too lazy & didn't have the space for another full-size PC). While the choice is ultimately up to you, I hope this helps!