There is the Blue Chaffinch (Fringilla teydea), a bird endemic to Teneriffe, probably you know. Once read it can be seen e.g. in the Parque Natural Corona Forestal (eastern-central area of the isle). Around Pico Teide it is very rocky, probably the coastal forests or in the northern parts there should be good places for birds. For your next time, central spain is a superb place for birding.
Go to southern Teneriffe (between Teneriffe and La Gomera) for whale watching (long finned pilot whale) and dolphins.
Have fun and enjoy the trip!