Driving yourself around Thailand isn't something I'd recommend, although traveling by car definitely has its advantages. The roads in Thailand, mainly due to the poor driving habits of a large proportion of Thais, are not very safe for those unfamiliar with the unwritten and haphazard rules of the road. However, the good news is that you can generally rent a car with a driver for the same price or less than you would pay for a car rental in North America or Europe. This makes a good option for a day trip to out of the way places.
Most the major international car rental companies have a presence in Thailand. However, most first class hotels will also have cars for hire and most major tourist destinations such as Chiang Mai have local rental agencies. Be sure to check out the car on offer before agreeing to rent it.
Driving yourself in Bangkok is neither recommended nor necessary. You can get to most of the sights by public transportation, and taxis are both cheap and plentiful. In provincial destinations such as Chiang Mai or Phang Nga, I personally would still discourage self-driving, but it is a reasonable option that many people have successfully exercised.