From experience a trip from the centre of Chiang Mai to Doi Suthep takes around 25 minutes. If you allow one hour for the round trip and another hour there for sightseeing you should be able to hire a red songthaew for around 400 THB for the two hours (make sure you bargain for the best price). Most drivers will point out highlights along the way and stop half way up the mountain at a viewing platform which overlooks the city. The drivers all wait in the car park while you go sightseeing. Don't worry they will spot you when you finally come back down from Doi Suthep.