Hi Joe,
Lanzarote is a stunning place for landscapes, I can't comment on birding locations though. Apart from the three main tourist resorts (Playa blanca, Puerto del Carmen and Costa Teguise) the place is pretty much all stunning / bizarre landscape, even the interior, mostly barren lava fields is photo-worthy.
The island is pretty small, you can drive from north to south in an hour or so, so as long as you have a driver, you can get to all the best spots.
For me, I'd say Famara Beach in the north, Papagayo Beach in the Sth East are the standouts beach-wise, though both attract lots of tourists, around Papagayo are a further series of beach inlets, that get less busy / not visited.
On the Sth West coast is El Golfo, a green coloured lagoon, that you ( in theory) can't get close to, but if approached from the south there is a way to walk down to it.
Just South of El Golfo are 'Los Hervideros' (means 'boiling pots') a series of rocky inlets where the sea crashes in spectacularly.
As a volcanic island, you might think of visiting Timanfaya, this is a national park of sorts, and the main area of volcanoes. However, a visit there consists of parking at a restaurant, and being herded onto a bus for a tour round the most interesting sights, you are not allowed to meander about on your own, so it's pretty useless photography wise. There are other volcanoes you can visit, less dramatic, but you can at least get closer on foot.
If you go there, or read up on Lanza you will come acroos mention of Manrique, and his influence as an artist and architect / planner is everywhere. Including 'Mirador del Rio' , which is essentially an observation point with a glorified cafe ! However it does give stunning views across to a small island off the north coast (La Gaciosa). Tho his buildings are tourist spots, rather than landscapes, they are stunning and striking, especially his house ( now the 'Cesar Manrique Foundation' and 'Jameos del agua' both worth a visit )
I'd suggest googling all the places suggested, and seeing what takes your fancy.
Hope you have a good time, and hope to see some of the photos here.
Me and Mrs Chap are going in Sept/Oct (having been a fair few times before) and I'm hoping to get some great shots myself.
with the above, good tip for Mirador Del Rio! The same for the "vineyards", weird little stone circles (which are also on some beaches too). I've not visited the Marina, that'll be on the list this time. 
