Bike Point in either El Medano, or Las Americas always seems to come out favorably.
Not sure on your best way to do Teide, but I do know you can stick a bike in the hold on the buses over there, so may be tempted to bus part of the way up, to get away from the coast/busier areas at least.
Resort-wise, places like El Medano/Los Abrigos/Golf del Sur are the smaller resorts, bit close to the airport for my liking, but a few bars/restaurants etc. Las Americas is the ‘big’ resort full of hotels and lacking in any character. Los Christianos is just around the coast, and is large, but has more a ‘local’ feel, still a lot of hotels/apartments, but the town centre/bay etc are nice, with plenty places to eat/drink. Never been to Los Gigantes.
If you fancy going off road, Lavatrax (Darran) is great, plenty of routes, and a great guy to ride with.