Yes, you can replace your PS4 hard drive. It's not the easiest process in the world, but it's not too shabby either. Given that every PS4 game requires hard drive installation, the 500GB drive included with the system may fill up quite quickly, so you may want to use this feature sooner rather than later.
Gamespot made a nifty new video showing off how to replace the PS4 hard drive. As you can see, it's quite easy. As a bonus, they replaced the hard drive with a SSD (solid state drive), which has the immediate benefit of making Resogun load twice as quickly. See for yourself: