Start early in the morning along the Corniche. First head to the Qatar National Museum. It opens at 09:00 and if you get there early you'll have the place nearly to yourself. Next head to the nearby Museum of Islamic Arts and check out exhibits spanning 1400 years of Islamic civilization. Stop for lunch at one of the many excellent seafood restaurants along the Corniche. Al Shaheen and Al Bander are both excellent choices for lunch.
After lunch take the boat ride from the Corniche to Palm Tree Island. Take a dip in the clear turquoise waters of the Persian Gulf. You can also enjoy a camel ride or water sports. Of course, in the summer you may want to skip all this! After a few hours head back to the mainland. The Weaponry Museum is open from three to six in the evenings. The museum features fascinating displays of armor and daggers.
Next head to the Souq Waqif, a traditional Qatari markets. The storefronts come to life in the evenings. Be ready to negotiate if you want to buy any souvenirs. The Villaggio is one of Doha's newest and trendiest shopping malls. Come here to get a feel for the luxurious lifestyle that many Qatari's lead. For dinner, head to Al Bateel for traditional Arabic fare, or to Ruby Wu's for innovative Chinese cuisine. These are two of Doha's best restaurants.