If you are not planning to leave the city, then taxis are fine for getting around, and provide good value for money. If on the other hand you want to tour the peninsula more extensively, then it is a simple matter to arrange car hire in Qatar, and the roads throughout the country are very good.
Dhows cruise around the coast, and it is possible to arrange a day trip as part of an organised tour quite easily.
For trips into he desert, a large 4WD with satellite navigation is essential, although if you are not experienced in driving in the conditions, then dune bashing is best left to an expert.