So generally how fast should I drive (just patrolling?) The biggest issue I have is if a road is curving i try to adjust to the curve but my car goes too far to the right and left (so my car is always moving right.left as I'm trying to control it.) to go straight.

Just follow the flow of the traffic (match their speed) or you can use an Xbox 360 Controller for Windows to make driving smoother. 

Learn to coast, you don't always have to be on the gas while you're driving. Same thing goes in real life, for that matter. Practice makes perfect, though. I can drive the CVPI in that game better than most of my friends can drive the faster cars like Lambo's and Ferrari's. They go fast but they lose control and don't know how to take corners or use the hand brake properly.

If you're using a keyboard, tap the keys to make small adjustments. Like unr3al said, you don't need to hold down on the gas, so just tap that (or hold in short burts) when needed.

Just practise I guess, within a week I managed to drive rather well.

I always tap the keys to control driving, I never go full blast on the gas, not even when responding (unless I am on the motorway or need to catch up speed). It feels a bit awkward at first, but you will get the hang of it, just like driving in real life. There are handling mods out there for modded vehicles that make them more realistic to drive (and IMO easier, they dont rocket back and forth and blast away giving you a minimum of 25 degrees turning each time you tap A or D etc.)

 

Or buy an Xbox 360 controller.

When turning, press the arrow key really soft, just enough for it to turn. I usually tap it like 10 times when making a turn.

I've found, that if you brake while turning, you're going to have a bad time.

Depends. If you're trying to drift around a corner, the hand brake is fine, and tapping the brake is okay if you're just trying to maintain your line during a corner.

The brake is nice in pursuits, and handbrake is good for rapidly turning around.