The Internet Browser on the PlayStation 4 is nowhere near as good as a laptops would be, but it's definitely 10x faster than the PS3's Internet Browser.
It's major flaw right now is that it doesn't have a flash player to play videos, so you will have to wait for the PS4 Youtube app before you can use that.
But if you just want to look up a game guide or a forum for some information on a game, it's pretty fast. In game just tap the PS Button, which will instantaneously take you out of the game and onto the main menu, then scroll over to the Internet Icon, then click on it ( it boots pretty fast ). After you're finished with it, close the Internet Browser, and click back on your game, which will bring you instantaneously back into the game. Overall the PS4's User Interface is very smooth and seamless, although it has less features than the Xbox One ( which too me is probably a positive attribute IMO / less features means less things will be buggy, and there is no slow down for the OS ).
Also background loading has to be the best thing so far about the PS4 ( no more long waiting times for downloads and installs ), and downloading and installing happens at the same time now. Plus quick installs for games ( either from PSN or from a Blu Ray disk ) which then keep installing while you begin playing the game. And loading times on games are now much shorter, especially if you have a "Solid State Drive" installed ( which reduces load times by almost half again ).
PS ~ A lot of gamers have not been utilizing the maximum graphical output that the PS4 has to offer. Here's a quick tip to maximize the image quality for your PS4 games.
Go to Settings in the Upper Menu > Then to the Sound and Screen Menu > Then Video Output Settings > Then change the RGB Output to "Full" instead of "Automatic". After that the screen will go black and reboot with a crisper & higher quality picture.
Edit: Most new modern TV's will be able to handle this setting, while older models ( 5 years or older ) might not. But this should result in a better image quality for your PS4 games. Also make sure that your TV's RGB Output is on the same setting.
Edit 2: I'll bet any money that the graphics comparison videos that you see on some of the Game News Websites ( IGN/ Gamespot etc. ) have not been utilizing the best graphical output settings for both Consoles ( X1 & PS4 ).
But this might not matter anyway because of the compression of the picture in these videos ( meaning reduced picture quality ). You really can't tell the difference until you've seen the graphical quality in person.
The PlayStation 4 games I have played so far are COD Ghosts and Assassin's Creed 4 ( I'm still waiting on Killzone: Shadow Fall & Battlefield 4 ), and these look unbelievably good on the PS4, plus they have an ultra-smooth Frame-Rate. I really didn't think that the PS4 games were going to look this good, but I've been pleasantly surprised with the quality I've seen so far. Future games can only get better from here on out, and I can't wait to see what my PS4 will do with all the great games coming out next year.