I like to think this game has two states of a drift.
1) A shallow slide where the rear has lost enough grip to oversteer, but is still controllable
2) A slide with an angle so sharp you can't undo it.
I remember I used to hit the latter state way too often, and only took hours to practice to get the hang of being able to initiate a shallow drift without whipping the car into it too hard.
If you're familiar with the common techniques such as the Scandinavian flick, brake, power over, lift-off oversteer, you'll find something comfortable with the car you use. Naturally all cars will handle and respond to the road and your input differently, so you'll have to play around with each technique to find what works best for you.
If you're doing Automatic, just note that if the car ends up downshifting to a gear too low, you'll know when the the RPM hits the limiter and you're getting no power through the turn. Before I switched to manual, I usually end up triggering this event if I let off the gas for too long in a slide. To prevent it from downshifting too low, you can pump the throttle.
Also, don't be afraid to let off the throttle for a split second if you anticipate the car will whip out of control (usually from landing after a hitting a bump). And if you find that the car is veering too deep of an angle when doing a lift-off oversteer, get back on the gas (not applicable for RWD).