How does an amoeba move?
It has a cytoskeleton, a network of fibers stretched across the cell which it can shorten and expand (by adding or subtracting actin monomers which build them) in a coordinated manner to more-less arbitrary change shape.

According to this paper, on surfaces they generally can adhere to it using membrane protein structures, thus the locomotion occurs due to friction.

To swim, they generate bumps near the front end of the cell and move them towards the back, similarly to how a swimmer uses hands.