If you are playing multiple games in a row against the computer, Reversi allows you to automatically alternate who will start the next game between you and the computer.
When configuring a new game, have a look to the first button with a down arrow; it should either be labelled Color: Dark, Color: Light or Two players. In the latter case, Reversi is configured to start a two-player game; if so, first, configure a game against the computer.
Select the button. In the menu that appears, there is an entry called Alternate who starts.
If the Alternate who starts option is disabled, every new game will be started by the same player: you will be the first player if the label of the button is Color: Dark, and it will be the computer if the label is Color: Light; to change who starts, see how to configure a game against the computer.
If the Alternate who starts option is enabled, then at each new game, who will be the first player will be automatically changed. So if you started a game, the next time you configure a new game, the button will be labelled Color: Light, and the computer will be starting the following game.
To enable or disable the option, just select the Alternate who starts entry.