Where do I find game modes and how do I start games in them?

The absolute best place to find SWEPS, Models, Maps, Tools, and Game Modes is garrysmod.org.

To get a gamemode a decent place to go is the Steam Workshop due to the one-click installs and uninstalls. To play a gamemode in singleplayer (or if you port forward, online multiplayer) go to SINGLEPLAYER, select the number of players and the map, then in the bottom right corner there is a box saying "SANDBOX". Click it and select the gamemode you installed.

If you want to play on a server that has a gamemode, just join. Gamemode files will be downloaded to your computer along with any custom content they have. Please note that once you join a server, you will not need to redownload their content. If they have addons that other servers have, you will not have to download these addons on those servers.