Hire a bike from Free Motion. They are somewhere in Playa del Ingles, I got a road bike from them – Cannondale super 6 and rode it up the mountain – it was very good (and hot).
OP: My wife is from las Palmas so have been to the capital loads. Las Cateras is a nice beach, 6km long and not touristy as it is really set up for the locals. If you want really quite and unspoilt then head up into the mountains, but you will be isolated from stuff to do and it takes a surprising amount of time to drive down the twisty roads. Mogan is nice as is Agaete in the NW, but Agaete is very quite (apart from the ferry to Tenerrife).
Maspolomas (lots of pigeons is the translation) is developed but the lighthouse end (faro) is nice. Playa Del Ingles (Beach of the English) is pretty much how you would expect it to be given its name!