I alternate between Tenerife and Gran Canaria in mid-January: I like climbing so both suit me well! Usually warm and dry and not especially windy. I've only stayed on the south coast of both so I can't comment on rides from the north.
Tenerife is dominated by the big ride up El Teide but it's well worth it, either from Los Cristianos or Chío. Going part way to Vilaflor from LC is relatively easy and you still get to descend from 1400m to sea level coming back. Masca is a nice ride too. It is busy in the resorts and it can be a bit of a PITA negotiating the roads, but once you're out of there it's grand. I once stayed in El Médano at the other side of the airport but I don't recommend it as a cycling base, mainly due to the blustery descent into the resort.
Gran Canaria has more variety so you don't have to ride to the top, but you should definitely try it. There are some easier valleys near Maspalomas, or take the empty coast road to Puerto Rico, catch the ferry to Puerto Mogán and ride up Serenity (see photo). I had one properly wet ride this year, my first one ever in the Canaries in January.
I did once try cycling in Fuerteventura, in July 2016: much too windy for my liking.
