It’s surprisingly easy to get to the Isle of Wight from London without a car, and you can buy a train ticket all the way from London Waterloo to Shanklin station, which includes travel on Wightlink’s FastCat. Take the train from Waterloo, then a short walk from the platform at Portsmouth Harbour down to the Wightlink catamaran. Around 20 minutes later you’ll arrive a Ryde Pier Head where you can catch a connecting train (which used to be an old London Underground train) taking you to Shanklin, via Ryde town, Brading, Lake and Sandown.
All quite possible to do in a weekend. The only thing to watch out for is the train through to Shanklin stops fairly early, to be sure of catching it, you need to be on the ferry which leaves Portsmouth at 9.20pm, although buses and taxis are of course available in Ryde.