Arcade and Gran Turismo modes are the order of the day. To progress in the Arcade mode, it's a case of finishing in first place. After you finish first on six courses in easy, more courses appear. Doing the same in normal and hard unlocks more cars. There are pleeeeeeeenty of courses to unlock, most of which are great fun to play on.
Stages vary from forest, to dirt track to city centres in appearance.
The San Francisco and Tokyo based courses are my personal favourites, as they do go some way to catching the atmosphere of both the cities. Well done, Polyphony Digital. The Gran Turismo mode is where things really hot up though. Eventually, you'll start getting decent cars and being able to compete in high-power races. Soon, you'll be king of the racing world.