If you don't want to worry about when to shift gears, the SRAM PC-1 chain is precisely what you need. This step riveted, 1/8" chain is designed to work with all single speed drivetrains and is not only incredibly strong but also inexpensive. Ditch the zero and get with the hero.
Includes 3-piece snap-lock connector
Compatible with single speeds
In order to install a new chain, it must be sized properly first. That requires a chain-tool, which are listed here. If you need to remove the chain, that can be done by hand or with a special tool, like the MLP-1.