DC / AC Inverter Principle of Operation
WARNING - projects like this are dangerous to beginners and the foolish

Power integrations publish many designs like this one: -



OK it's a single 24V DC output but the transformer can be re-wound (yes nearly all these designs require a hand wound transformer) for two secondaries of 12V each.

Premier Magnetics (who supply the pre-wound transformers) are also quite keen to supply design notes that cover flyback applications and designs. Designs include power integrations, TI and fairchild chips and look good but I'm not a novice and there may be several things that can confuse a beginner. Here's a premier magnetic design that also uses the top-switch chip: -



I would certainly consider either of these two companys' offerings. Here's another one that is closer to your requirements: -