There is plenty to see and do without a car in Chiang Mai, plus parking can be an problem especially if you are staying in the central parts of town. If you are simply spending a couple of days exploring town and the immediate surroundings, I would not advise renting a car.
If you have a more time and want to head out for a few days to explore more of northern Thailand, then perhaps a rental car would be good.
If you do rent, a couple of things to remember ... in Thailand we drive on the left side of the road (like the UK) ... there is a lot of give and take on the roads when it comes to right of way and road rules ... if your drivers license is not in English you need to get an IDP (International Driving permit) from home.
There are plenty of decent car rental companies in Chiang Mai. All of the global brands are here, plus you have long time local companies like North Wheels, Pop Rent a Car.
As with any rental company, inspect the car in the presence of rental agent and have them mark any damage on the rental form. If you are nervous, take photos of any existing damage in the presence of the agent (maybe even work them into one of the shots).