How do I start a game I've created in Rec Room?

You'll need a "Game Rules" chip and either a scoreboard or a "Set Game State" chip.

The scoreboard projector has a button that players can use to start the game. The "Set Game State" chip can be used to start the game from a circuit signal. From the wiki:

A signal unequal to 0 (True) on the red pin starts the game (or more precisely: it starts the pre-game sequence and after 5 to 6 seconds, it starts the game). A signal unequal to 0 (True) on the green pin stops the game.

The game rules chip can be configured using the configure tool on your makerpen to set up all sorts of settings for your game, e.g. how many teams there are, how many players per team, respawn mechanics, etc. You need a game rules chip in your room for there to be a game that can be started.