Honestly, exploring the city on foot is pretty fun. I did a lot of my early game traveling on foot, just soaking everything in. You see a lot of neat details you would otherwise miss by driving.
As you explore more of the map, there are fast travel waypoints that make travel easy.
Some cars handle better than others, and eventually you get somewhat accustomed to it. Motorcycles have the best handling in the game though, and I prefer to use them over cars.
Enjoy the game. It's got it's flaws but imho it's great.