First of all, welcome to Doha (although I am not there now!)...domhow said:
Anyways, your first and primary objective is to take ALOT of fine photos and post 'em here!
Then comes, the chillin' out part
You can go to the malls, City Centre, Landmark, Hyatt Plaza...etc
You can also go and stroll by the beautiful corniche...
You can go to the many parks
You can also go to the souk beside the (ta3riff belislam), which is the building with the spiral minaret.
You can rent a 4x4 and have some nice off road action.
You can also eat at that road (dunno its name) with a $hitload of restaurants.
And...
You can take more photos for us!
Anyway, maybe other qataris here could tell you more things to do and more places to go.
Have a nice stay