October is still very warm with plenty of sunshine.
Areas to stay - Las Palmas, 20 mins drive from airport, very built up and suburban, some places of interest, but very local in that it's a lived in city and tourists usually visit it rather than stay there for any length of time.
Puerto de Mogan, 50 mins drive from airport, very small resort but very picturesque with a small beach, large local market on a Friday and lots of small cafes, bars and restaurants. It's the furthest resort in the South but well worth a visit if you're not actually staying there.
Puerto Rico - 40 mins drive from airport, very British, lots of bars, very lively with 2 beaches, one in the town itself and then 5 mins drive away is Amadores Beach (my favourite on the island).
Melonares - 30 mins drive from airport, 5 star hotels, boutique shopping, promenade and very close to Maspalomas sand dunes and beach.
Playa del Ingles - 20 mins drive from airport, biggest and most commercial resort, many bars, clubs, hotels.
I would recommend a trip called Island Delights, it's a tour organised by Thomson/First Choice and takes you around the island in an 8 seater air conditioned people carrier. It goes right into the "Gran Canyon" of the island, makes regular stops, gives lots of information and lunch in included, well worth doing to see the beautiful inside of the island, away from the commercial areas. I believe they also do an evening tour as well now so you get to see the sunset - and on a good day you can clearly see Mount Tede on Tenerife.
You can book the trip with a Thomson/First Choice rep in any of the hotels they operate in or go to their website and pre-book before you go.
Other trips are availble, just a word of caution though, please book any trips from official companies as they are you are fully insured and covered - I'd also recommend a boat trip to see the coastline - you can get a water taxi from Puerto Rico into Puerto de Mogan (one way or return) - and perfect for the Friday Mrket, so early though, it gets very busy by 12 noon.
Hope that helps.