How to Play Quake III Arena

Get the game data: Quake III Arena on Steam or GOG, or the original CD.
Install a free modern engine, either ioquake3 or the faster Quake3e, and point it at your baseq3 files. Stock Quake III asks master.quake3arena.com, which lists around 945 servers, so your Steam copy already finds plenty of games. The ports are worth it for being actively maintained, widescreen, and free of the old frame-rate ceiling - and they also query directory.ioquake3.org (about 890 more, heavily overlapping) so you see the union of both.
Staying on the stock game? Open the console with ~ and enter sv_master1 "directory.ioquake3.org", then refresh the Internet list.
Playing CPMA / Promode? That scene uses its own engine, CNQ3.

Joining a server Copy the connect string, open the console with ~, and type /connect ip:port.