Just like everything else in woodworking, shaping your piece can be done with a number of different tools.
Generally, I rough-cut pieces on a bandsaw and then finalize the shape with chisels, gouges, spokeshaves, rasps, files, sandpaper wrapped around dowels, and a lot of time.
I haven't found a quick way to shape pieces. An oscillating spindle sander can come in handy, but you wouldn't be able to reach tight corners and you'd still need to do final sanding.
However, just like sanding, you can speed up the process by starting with your most aggressive tools—coarse rasps and files, deep cuts with chisels and gouges—to remove the bulk of the waste.
Working under a raking light reveals parts that need more attention, and using a square to check that your edges are perpendicular to the faces ensures that both sides of your piece will be identical.
After I shaped the legs, I trimmed the tenons that pass through the seat. At this point you have a workable stool.