First, make sure you are allowed to leave the airport with your passport/visa.
Visa on arrival is available for many countries, but costs US$28 for 30 days. But that's only certain countries, other ones are more complicated.
Wikitravel is limited, but suggests the following sites:
The Museum of Islamic Art, Doha
Souq Waqif : the traditional old marketplace of Qatar. Has many good restaurants, especially at night time. Also sells many national products - bargaining is recommended.
The Pearl : a man-made island connected to Doha by a bridge. You can find a big variety of restaurants and shops, mainly in the high range.
Villaggio Mall: a spectacular Venetian style shopping mall with a canal and gondolas as well. A huge variety of shops from casual to luxury.
Mathaf : The arab museum of modern art
Katara : Cultural village which is home to many international and Arab restaurants, a beautiful beach, and holds many cultural events. Definitely a place to see.
Aqua park : Aquatic Funfair.
Personally I'd see maybe one thing, do it properly, and then go hunt out strange foods and experiences. Middle Eastern cuisine is everywhere, and in many forms — kebabs, breads, hummus, the list goes on.
Don't be afraid to venture into the Souqs looking for a meal; it will
be a unique experience in an authentic setting, and although some of
the places you see may look rundown, that's just the area in general,
and the food will be probably be quite good. Be advised that many of
the restaurants in the Souqs (as well as the shops) shut down during
the afternoon hours. If you are in a funny kind of mood, you can try a
McArabia—McDonald's Middle Eastern sandwich available only in the
Another thing that I did in Dubai, and apparently they offer in Qatar as well is dune-bashing - riding around in big Landcruisers across the desert - it was AMAZING fun! However, after my suggestion of it to a relative went bad - don't do it if you get at all carsick ;)