Quake 3 is a multiplayer only game, as PhilsComputerLab says. You can play it on your own, against it's very good bots (bots are computer controlled oponnents, programmed to play the game as humans would), but you're still playing a multiplayer game, just without any human opponents.
Most first person shooters also have a single player campaign, including Quake 1, 2, and 4, but I personally don't like any Quake game other than 3.
FPS (first person shooter) campaigns vary from game to game, some, such as Doom 1 and 2, or Quake 1, or are mostly simple move from the beginning of the level to the end, killing everything in your way, and perhaps finding a key to unlock/open a door. Other great games like this are Duke Nukem 3D, Blood, Unreal, No One Lives Forever, , Halo 1 and 2, Half-Life 1 and 2, and lots of others.
Then you have games like Deus Ex, No One Lives Forever 2, and sadly not too many others, that require a bit more brain-work, requiring you to upgrade your weapon skills with points you collect from exploring and performing in-game objectives, etc.
You have the realistic shooters like Operation Flashpoint: Cold War Crisis, which is the only one I can name, since it's the only one I've played - I was put off as it's too realistic, i.e. too hard. It's so realistic that, for example, when you see an explosion in the distance, the sound comes later since (as in real life) in the game sound travels much slower than light. You can drive or pilot almost any vehicle, you can apparently lose the game but your side (you play a soldier in the army, in a war zone) can still win, and it's an incredible game. I've tried it several times over the years, and even though it looks rubbish (even when it came out, it looked rubbish, presumably so the majority of the PC's time could be spent calculatin all of the in-game physics, all of the enemy and friendly soldier's actions, etc) it really is an amazing achievement. And lots of dedicated fans have created playable levels and battles that you can download for free and play. But to me it's too hard to be fun, I mean, in the first mission, you have to get to a car in a small village, and drive to a given locations. Sounds easy, but the village has various snipers positioned to look for intruders, and so you have to crawl through bushes, around corners, etc, stay as much out of sight as possible, and constantly search for snipers to either avoid or kill them - though if you can do kill a sniper then you must be careful not to give away your position. And since this is such a realistic game, one sniper bullet to your head and it's game over, even though you've no idea where the sniper was.
I've not played any other Operation Flashpoint games, so I don't know if they've been dumbed down like most other game series, nor have I tried the Arma games (the sort of successors to the Flashpoint games).
Then you have the games that are claimed to be realistic, but are nothing of the sort, such as the Call of Duty games. I don't like the CoD games, but that's nothing to do with their false realism claims (it's not realistic to be able to heal any injury just by hiding behind a wall and waiting a few seconds, it's not realistic to be able to swim when carrying two real life military rifles and a lot of ammunition, it's not realistic to be able to aim when running, it's not realistic to be able run backwards down a staircase, etc). I just don't find the Call of Duty games to be enjoyable.
Anyway, if you want a PC FPS that's simple, I'd say start with Doom, as it's pretty basic but very, very good. Then play Doom 2, Quake (I don't like it, but lots of people do), Blood, Duke Nukem 3D, Unreal, etc, and see what you like.
If you want something a bit more advanced, and one where the story is important, try Half-Life, No One lives Forever, or the utterly fantastic Deus Ex. If you try Deus Ex (the first one, from 2001, not the two sequels), then stick with it, it's difficult at first, since you have almost no weapon skills, and don't yet know the game's mechanics. It's well worth persevering, though, as many people consider it to be the best PC game of all time.
Basically, there's lots of great PC FPSs, if you like the genre then you're set for many years of fun!
