There is now a supertram service to La Laguna, the older capital just a few miles away. It's well worth the half hour journey. Trams run at least every 15 minutes. Buy your ticket from a machine at the tram stop (press the button with a Union Jack to get instructions in English). It's flat fare system, it cost us €2.30 for a return journey. Take the tram marked La Trinidad. Stay on to the end of the line of the uphill journey. The terminus is in a nondescript street (though I had a superb kebab in a restaurant there), but if you walk forward and go left at the top you will be into the old city. The cathedral was closed for repairs, but not far away is the lovely old church of the Conception. (NB Remember that just about everything is closed in the afternoon for siesta, including churches). The cafe Don Tito can be recommended, especially if you like crepes!
In early December though it was cold, 5 or 6 degrees only. We think that's unusual: on local TV they were getting very excited about the snow in the hills not far away.