How did the Shuttle do it?
While I don't know exactly how Shuttle did it, there are essentially 2 ways to do it:
* use membranes (not so easy with cryogenic fuels, but used for Kerosine/peroxide, etc.) with pressurized volume on the other side. Notice that the membrane doesn't have to be elastic, just flexible.
* use the gravity/inertia to locate the fuel around the collection points.
The latter is used for most of the fuels/oxidizers. Positioning of the liquids in the low gravity can be done by either starting a small engine or by slowly extracting some amount of fuel/LOX into a small pressurized volume (electric-driven pump is enough) and starting the engine using this collected fuel. This stage is usually called "pressurizing" and we can hear pressurization reports/calls on some footages. Then the started engine pushes the rocket one way, the fuel moves the other way, and this is where the collector will take it from. This moment is critical, since the fuel tends to slosh around a lot when Gs change quickly. So, you'll see lots of buffles or other interesting geometry usually around collector ports.
How can the electric pumps collect fuel/LOX, one might ask? Surface tension. Over time (and it's pretty short period of time), when blobs of fuel/LOX stops flying around in zero Gs, they merge and/or hit the walls of the tank. Then they cover the walls in a pretty even layer of liquid. Notice that the fuel blobs are all over the place and the collector port is essentially dry at 26:11, but they're fewer of them and the collector port is covered with liquid at 31:27. That's surface tension. It allows the pumps to extract enough to start the engine and push the rest to the collection port.