Or would we hit the surface and stop there?
I think you are confused as to what the 'surface' of Jupiter or Saturn are. They have a large liquid hydrogen centre, but this is surrounded by an incredibly thick layer of atmosphere, which has clouds, gases, liquids etc.
So you would first pass through the outer layers of atmosphere, falling through denser and denser gas until you float at a height which matches your density. Unfortunately, the pressure would have crushed you by this stage, so you wouldn't be able to enjoy it.
The core would still be far below you.
If you like science-fiction, try A Meeting with Medusa by Arthur C Clarke, an awesome writer who uses hard science in his stories.
You wouldn't be able to pass through. Yes if you were to go down far enough you'd be able to find a height that goes with your density but you'd be smashed together from pressure.