I dont see Qatar as a tourist destination nor a haven for tourist attractions, however here are some of the places that you may want to visit:

1. Qatar Museum
2. Katara Village
3. Corniche
4. Aqua Park
5. Qatar Zoo
6. Beaches like Sealine, Dukhan, Zikreet etc..

I've been to those places, and they are just fine. Good places but I dont know, I feel like it's lacking something.. oh well, go and see for yourself.

Have fun!