I was in Gran Canaria in June 2003, to be honest there isn't a huge range of birds to see but still plenty of nice things to keep your interest. First off, there are plenty of escaped birds, glossy starling species and bulbuls etc often get the pulse going for a bit trying to work out what they are. As for 'proper' birds, Plain Swift is often abundant in the mountains, Sardinian Warblers, Spanish Sparrows and Canary Islands Chiffchaff can be found around gardens, along with feral Monk Parakeets and Waxbills. Canaries are often in scrubby and rocky areas. The Oasis de Maspalomas is good if you can stand all the people, I had Whimbrel, Ringed, Little Ringed and Kentish Plover, Turnstone and Collared Pratincole here, along with Moorhens. Cory's Shearwater offshore. The golf course nearby had 23 Hoopoes on one hole and a few Southern Grey Shrikes.
There are a few Blue Chaffinches in the pine forests, but they are rare and I didn't see any, I did see plenty of Canaries and Turtle Doves though, and the scenery was spectacular.
Also look out for the Gran Canarian Giant Lizard, some of the ones I saw were well over two foot long with the tail, look in sunny roadside areas behind chainlink fences, they know they're safe there.
If, as you say you're new to birding then there are plenty of new and exciting things to see without being overwhelmed by unfamiliar species. In a fortnight I managed a list of about 45 species only. Have a good trip.