If you have never played Quake 3 Arena, grab the demo and check it out! You can not play custom maps or mods with the demo.
(For the full version of Q3A) Be sure you install the Point Release first, then copy the updated game engine and replace older version.
The PAK files only version can be used instead of the Official Point Release 1.32 and is only suitable if you are going to use an alternative game engine like ioquake3 or Quake3e.
Start Quake 3 Arena and bring down the console. For most keyboards this is the done with the "`" key, which is left of the "1". Type the following, replacing 1920 and 1080 with your screen size width and height. This step may not be required with the Quake3e engine.
\r_mode -1 \r_customwidth 1920 \r_customheight 1080 \r_fullscreen 1 \vid_restart
Depending on the game engine you are using, there could be a heap more options you can use the improve the visual quality.
Q3A was released when games ran at 800x600 pixels, which is tiny by today's standards. These two mods help to improve the visual quality.
There are so many great maps on ..::LvL, but if you are after a quick start, grab the ..::LvL Q3A - 20 years map pack.
If you are unsure what to do with a downloaded map, check the How to play a Q3A map FAQ.