have you taken a look at the videos reviewing these consoles and the games they play? They all look fantastic, we've left the "old" graphics realm for more than a decade now, anything that runs on the PS3 will already look amazing.
that being said, if you have a budget of 600 bucks go for a PS4, once they're in stock again a bundle will cost maybe 400-450 bucks at your local target/walmart/best buy/random store. Then you'll still have 150-200 left to get a bunch of games (or save up for upcoming games, some of which look amazing like Watch Dogs, Destiny, Dragon Age: Inquisition, Final Fantasy 15, etc)
If you build a PC you can get a nice mid ranged PC for 600 bucks, but that requires some things that you may or may not already have like a monitor, keyboard, mouse, and headset/speakers, all of which take quality money away from the actual PC itself, so you'll be left with something that won't run the latest and greatest games. And you certainly won't be able to play those console exclusives (not a problem for some people)
If you just want to game, then a console is the way to go, I recommend a PS4 but both it and the Xbox One are fantastic.