Q Chain Reaction are pretty serious bikers doing high speed longish distance road rides.
A new riding group is forming - check out Qatar Cycling Community on FaceBook if you are interested. Based on MTBs - one part looking for off-road fairly long rides. The other more of a cruising section.

There is a dedicated cycling track alongside the Lusail Development. 2kms long. There is also one almost complete running down alongside the highway between the old and the new Airports. Approx 10kms long when fully opened. The southern 8kms is useable now with minimal obstructions. This track is fully lit. Park at the Old Marriot Hotel and cross at the nearest roundabout to find it.

Cycling on the Corniche is probably still illegal - there are signs banning it - but some do it. Most of The Pearl is pretty traffic free and there are also pathways/cycle tracks being installed there now as part of the landscaping and planting programmes.