Muhammed Bin Abdul Wahab Mosque
Also called State Grande Mosque. Beautiful huge mosque where i got to visit in full solitude. Separate entrance and sections for men and women, women given capes and scarves at the entrance.
My favourite part of Doha was walking along the 7km long Al Corniche early in the morning when there were only couple of joggers around. Stunning sunrise. I've heard its beautiful also at night time.
If you want to see camels, this is the place! A little camel farm right next to the Souq.
Museum of Islamic Art (MIA) (متحف الفن الإسلامي)
Beautiful building at a beautiful spot. The museum displays beautiful old arabic pottery, jewelry, carpets etc. Free to enter.
The shops along the small alleys of the bazaar are open from 8am-12, 4pm->. To avoid crowds come early, to see the crowds come late in the evenings.
Hamad International Airport (DOH) (مطار حمد الدولي)
Full of shops, modern, well functioning airport near city. Get there with taxi (affordable) or bus 777 which leaves every 20min and goes to Souq, MIA, corniche, pearl etc.
For local data SIM card, go to Ooredoo at arrivals (turn right after customs).
Al Mourjan Business Lounge
Huge and super impressive lounge by Qatar Airways. Normally only for business and first class passengers (one world emerald doesn't normally even get you in), but I think I got in due to morning rush hour...