The main screen of Spore gives an overview of the galaxy. Several stars are highlighted, and each one can be the basis for a new game, with a new race. To begin a new game, click on a star, and follow through the process:
Note that it's a good idea to start nearer the centre of the galaxy, as the ultimate goal is to reach the centre. Players starting their game further away from the centre will not be too disadvantaged, though, as wormholes are available as shortcuts between different areas of the galaxy.