Goals in GTAIII
1. Stay alive
2. Make money
3. Get revenge

While these goals may sound very simple, as you play the game they become more and more difficult.
As you complete jobs, your employers will introduce you to more people who will require your services.  Each employer has specific needs and tasks that must be accomplished. 

For example, your first employer Luigi owns a strip club.  He will need you to drive girls and pick them up.  The second employer is Joey, a mechanic.  He will need you to pick cars and deliver them to other people or places. 

There are also several other ways to make money, such as vigilante missions.  These involve finding certain vehicles and assuming their role.  For example, activating a mission while in an ambulance will require you to pick up injured people and get them safely to the hospital.  Other missions include, taxi cab driver, fireman, and police officer.

The non-linear design of the game is one of it's strongest features.  Players choose their own missions and when they wish to play them.  This gives a great deal of freedom to the player making the game more fun and realistic.