Gay nightlife (and anything specific by day really too) centres around Playa del Inglés.
There's a few options in the capital Las Palmas though it's a long way to go for a bit of fun if you're staying in the south with the fun on your doorstep. Las Palmas is a nice alternative though if you like to mix the fun with some real Spanish/Canarian culture and atmosphere as you won't find much of that in the Yumbo or the south. Las Palmas, in addition to several gay bars, clubs, saunas..... also offers visitors museums, galleries, theatre, opera, classical and popular concerts, various major events throughout the year (second largest attended carnival in the world) and of course most important of all, shopping to die for at prices lower than the duty free in the airport.
If you want to be in the heart of everything though and do little more than relax on the beach all day & party by night then look for somewhere in Playa del Inglés (near the Yumbo center)
THE SOUTH - By day, the place to go is the sand dunes. The dunes themselves have an area to the south which are covered in bushes. It's in and around these bushes that gay sun seekers tend to favour setting up camp for the day (I wonder why.......)
On to the beach itself which seems to be split into three areas. Nearest to the lighthouse is where you'll find the families. A ten minute stroll further along the beach brings you to the nudist area (birthday suits are not compulsory) so this is where everyone goes (by beach bar number 7). Past there is a third area, again popular with nudists but also with more families as you're reaching the end of the beach and into Playa del Inglés. So, stick to the bit in the middle next to beach bar number 7. The easiest way to get onto the beach is drive/taxi to the Annexo II commercial center, and walk along the beachfront from there or drive/taxi to the lighthouse "Faro de Maspalomas" and do the same from that end. A fun way to get on the beach (lots of cruising on the way back) is go to the Riu Palace hotel at the end of Avenida de Tirajana in Playa del Inglés and trek through the dunes themselves. You can't get lost as a) you can see the sea and b) follow the crowds.
ENTERTAINMENT IN THE SOUTH - By night head for the Yumbo Centre, the main shopping/commercial centre in Playa del Inglés. Unless you want to get there in time for dinner (you can't miss the gay restaurants because they display the gay flag, though in and around the Yumbo literally everywhere is gay friendly) there's little point in arriving much before 10pm. On the lower level is a huge open-air area with all kinds of stalls and shops to have a look at for the early part of the evening. The biggest & best is Sparkles Show Bar on the first floor, There is also Ricky's Cabaret Bar on the ground floor. These bars tend to fill up from 10pm onwards and the shows start after 11pm. Different shows each night of the week and karaoke beforehand.
From early evening (after dinner) the terrace bars on the lower level fill up, from 11pm onwards a few people are in the rest of the bars but most go to see the shows or people-watch on the lower terraces. Not long after midnight, the rest of the disco/music bars start to fill up. The top floor bars (4th floor) are all located in a circle and people tend to work their way round, having a drink in each. The last bar you come to on the top level is also a nightclub and is open from 2am until around 5am or 6am.
The leather and "dress-code" bars which are the places to cruise are all at the other end of the center (1st and 2nd floor).
The Cita Center is a 15 minute walk from the Yumbo or 3 minutes in a cab. So, another destination by night to head for is the CITA center, basement level.
GAY EXCURSIONS - If you're looking for something to do, somewhere to go by day other than the beach then there are gay catamaran cruises & also a gay quad safari into the mountains.