I have only stayed in Puerto del Carmen, there is a walk from Arrecife back along the coast to Puerto Del Carmen, I don't know about bird watching but plane spotting is guaranteed on this walk as it passes right next to the airport, you can stand under the flight path as the plane comes into land, they are not that far above you, we usually get the service bus into Arrecife and walk back, I think its about 10k.
A shorter walk the other way from Puerto Del Carmen brings you to Puerto Calero, with a boat trip back if you like, this is probably a better walk for bird watching.
I have toured the island by car and bus, its very dry, there are probably some good walks in the hills but I havn't done any.