I've hired bikes in Tenerife a few times. I was most recently there 3 weeks ago and got a Merida from Freemotion; 2 years ago it was a Bianchi from Bike Experience. I can recommend them both but Bike Experience get an extra nod after coming out to rescue me when I got my second ever puncture in the Canaries on the same ride (and I've not had one since of course). I think they do single day bookings too.
The climb up Teide from Los Cristianos is great; the descent from the top of the caldera to Boca Tauce is a welcome relief after the long ride. If you need water, stock up in Vilaflor as there's nowhere else en route.
Masca is also a great climb but it is usually spoilt by cars and buses getting stuck on the many hairpins and blocking the road.
If you want to go to Masca or up Teide from ChĆo, you'll need to negotiate getting from Los Cristianos through Las Americas and Costa Adeje which is "fun" to say the least.
Finally, if you fancy the challenge of a really steep climb, try Camino La Fuente in Valle de San Lorenzo.