there is a bus service for the airport to Los Cristianos and Playa de Las Americas. I can't remember how much it costs, but I think it runs at least every hour. Like royclark says, though, a taxi will set you back less than 20 Euros and should only take around 20 minutes.
As for things to do/ places to go etc, it depends on how far you want to travel. There are plenty of bars/restaurants in resort offering Karaoke/Cabaret, with Las Americas offering the same plus a few more 'noisy' bars for drinking and dancing.
Outside of the main resorts, the island is a totally different place, and I would highly recommend going up to the Ca�adas national park at the top of the mountain - It's a totally alien world up there and where they filmed the original planet of the apes. If you can get a trip up there vian Masca, this offers some truly spectacular natural scenery down the baranco.
When are you planning on going, because traditionally February is very quiet and a lot of places take a couple of weeks off. Also there is often a bit of dodgy weather. Still much nicer than UK though.
I remember finding it very starnge that I could drive 30 minutes from the 85-90 deg heat on the beach and go and have a snowball fight in Villaflor, then drive back down the mountain and sit in a bar sweating with an ice cold beer. Throwing snowballs at people from a cool box tends to confuse them a bit :-)