The evolution of GTAIII is interesting. With each game, the game play got better. I enjoyed GTAIII when I first played it, but I could never go back to it. It was maddening to do those police mission where I had to constantly drive without seeing where I'm going just so I could shoot the car next to me. I was able to offset that a bit by using the rocket launcher sometimes.
In Vice City, I still had the awkward side-camera shootings, but I had something else to help me out. If I took the unmarked car, then I could keep my wanted level down by visiting Spray and Pay often. I don't believe that Vice City introduced the police motorcycle. At least I don't remember doing that. In San Andreas, those police missions became a lot more open for me when I got to use a motorcycle.
In general, the addition of motorcycles in Vice City was great. It became more than just driving with awkward shooting. You could combine driving with shooting. The full driving-while-shooting experience didn't happen until GTAIV. That was nice, but it also changed how I drive. In GTAIII*, I could make slight adjustments in steering with the mouse while using the keyboard for harder turns. In GTAIV, the mouse doesn't affect steering at all, so I have to rely on keyboard commands.
I totally did not want to like the GTA series. I never played I and II. I wrinkled my nose at III because it sounded like one of those snuff games that'd get adolescents off. It's the same reason I snubbed Postal. But when I watched a friend of mine play, I said, "You know, that looks pretty fun."