Do an escorted island tour, so you see it all - and make sure that it includes going up to see the hot geysers on the top of the island, really worth seeing.
you must see mirador del rio.its a crime to go to here and miss the view.its spectacular
the "secret" of lanzarote is good too
also not overly pushed is a cafe/restaurant on lz 10(road) with a fabulous view and even if you dont eat the view is beautiful.just drive in and buy an ice cream
also off the tourist trail is the fort at teguise.
i love the island