DH and I used to go to Tenerife every year between Christmas and New Year. One year we were in the North, which is a bit cloudier but not so touristy, and four other years we went to Costa Adeje. It was fabulous! Lovely out-of-season vibe, too. We avoided the town centre but there still some nice cafes about on the outskirts, although TBH, we rarely left the hotel to eat. We went on some nice walks along the beach and one year did a trip up the volcano, which was really good. For us, it was about chilling out together in the sun and the weather was perfect for that - not too hot but we were able to sunbathe a lot of the time! Highly recommended!
It's where they filmed the original Planet of the Apes, too. What more could you want from a holiday?!!
BTW, there is a water park and a wildlife park (poss same place, really not sure) that I've heard good things about but we never went there and I don't know whether they're open off-season. That was an amazingly helpful bit of info, wasn't it?!!
Yes, Mount Teide is the volcano. That all sounds wonderful - hope you book it and have a brilliant time! New year's eve is great there, too - even if you just stay at home in the villa you'd be able to see massive fireworks all over the island.
Beware, if you're looking for the sun, there is SNOW on the Teide in the winter, and most of the time, the half top of the mountain is covered by fog and clouds, so, I wouldn't recommend to rent something there in the winter. The Teide is 3500 mt high, so even in august it is very chilly up there. Have a look at costa adeje, very nice. Las Americas is not that bad, it really depends, where you go. If you can rent something here www.parquesantiago.com/es/apartados/parquesantiago4, you are in a very privileged place. The parque santiago 4 is really nice, first line on the sea, a lovely promenade. The nr 3 is ok, but not the nr. 5.