A good portion of Kenmore homes have their own swimming pools, so most people have a friend or neighbour where they can go for a dip, but I assume you mean public swimming pools. ;) The closest Council swimming pools are at Jindalee and Bellbowrie. Jindalee is slightly closer by distance, but the drive to Bellbowrie has no traffic, so they're roughly equi-distant by driving time. Brisbane City Council provides loads of information on the different facilities at each pool here: http://www.brisbane.qld.gov.au/facilities-recreation/sports-and-leisure/swimming-pools-in-brisbane/index.htm
I seem to get these questions via email, so will keep having a swing. Ha.
There is a pool at the school nearby the Kenmore Plaza on Moggil Road. If you are after serious training, the Carew swim school in Chapel Hill (next suburb over - also on Moggill Road) has a history of training swimming elite, like Kieren Perkins.