Here's a quick list of some pretty nice gameplay wise First Person Shooters (FPS) games that you can play on Windows, Mac, Linux.
They won't be visually gorgeous but you'll have lots of fun and the community in general is fantastic :)

Nexuiz

Nexuiz is a game in the spirit of Quake and believe it or not their engine, codenamed Darkplaces, is actually based on the venerable Quake 1 engine !

Don't be disappointed though, the engine was substantially modified and it looks pretty well with some nice effects.

It also features a single player campaign where you can get used with the controls and gameplay.

Resources:
Download | Screenshots & Videos

Sauerbraten (Cube 2)

Cube 2: Sauerbraten is a free multiplayer/singleplayer first person shooter and what makes it to stand out of the crowd is that it's engine was specifically designed to allow map/geometry editing to be done dynamically in-game.

Resources:
Download | Screenshots

Warsow

Warsow is another Quake 3 / Unreal Turnament inspired game with cartoonish / cellshader graphics focused on speed, trick jumps and the art of movement.

Warsow’s 3D OpenGL game engine is based on QFusion which is an enhanced version of the original Quake II engine.

Resources:
Download | Screenshots

OpenArena

What more can I say. This is a free reimplementation of Quake 3 with free content (models, sprites etc.)

Resources
Download | Screenshots

Tremulous

The only one in the list that isn't truly a Quake 3 like game but uses the Quake 3 engine.
It blends a team based FPS with elements of an RTS. Players can choose from 2 unique races, aliens and humans. Players on both teams are able to build working structures in-game like an RTS.

Resources
Download | Screenshots

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