How can I support a room expansion that will extend above a garage slab?
A diagram with dimensions would help.  But supporting the floor is the easy part - basically, you'll need to build a cripple wall, framed with 2x4 lumber, to support a rim joist that you'll hang your floor joists from.  Set the cripple wall on a pressure treated plate lagged to the floor with concrete anchors.  Floor joists will be sized with a span table  Note that your garage floor should be sloped from back to front - you'll have to accommodate that in your framing to end up with a level floor.

You'll also have to determine if the wall at the back of the garage that you're intending to remove is load bearing or not.  If so, that will be alot more work involving a licensed engineer, to figure out how to do that correctly.  You'll also have to figure out how to support the new floor at that side, so you'll need to understand the framing &amp; foundation of that wall.

Not sure where you're located, but most US building codes have specific rules about the walls between a garage and living space.  Things like type &amp; thickness of drywall, fire rated doors, sealed penetrations etc.  You'd do well to check the codes for your community before you go to far.