I would say the closest beach to The Gap is Nudgee Beach but agree with Rachael20, The Gap is a bushland area and not close to the beach at all. Bribie Island IS probably the closest, nicest beach, as the water at Nudgee tends to be a bit murky.
If you're looking for a suburb, specifically to be near to the beach, honestly, The Gap is probably not what you're looking for. The closest beach to The Gap, I would say is probably a toss up between Bribie Island and Redcliffe. I personally would head to Bribie Island! The two biggies, The Sunshine Coast and The Gold Coast are about 1 hour 20mins drive away. As far as places to go swimming, aside from potentially your neighbours pool, there are a couple of local pools in The Gap (i'm not sure if they're open to the public anymore...havn't been since I was a kid) but there are a number of pools in surrounding suburbs that wouldl fit the bill. Hope that helps. Like I said, if you're looking to be closer to the beach - you'll be further from the city, but I'd look further north or south as The Gap is bushland not beach.