Electrons can move at any speed up to, but not including, the speed of
light. In atoms, typical speeds relative to the nucleus are on the
order of a few percent of the speed of light.
The picture you hint at, with electrons gaining energy then
suddenly appearing at a new place is suggested by the common phrase
'quantum jump', but it doesn't accurately describe how they behave. The
electron states are sort of cloud-like. They neither are located at
particular places nor do they have sharply-defined velocities. The
'wave functions' describing these clouds can have surprising
properties, including being non-zero in two regions even though they
fall to zero in between. It wouldn't make sense to think of an electron
hopping between the two regions without going in between.
The quantum states change in time according to something like
Schrödinger's equation. It gives purely continuous changes, with no
leaps at all. That's true not just for electrons but for all
microscopic collections of particles.
There is a mystery to quantum mechanics, however. If we look at
what should come out of the Schrödinger equation for big collections of
particles, say like a cat, it can be states which are composed of parts
representing a live cat and a dead cat. We never see anything like
that, just one or the other. So it's tempting to think that somehow the
large-scale world does 'jump' into a particular state rather than just
obeying the Schrödinger equation. People argue about all sorts of
different ways to understand this issue.
(republished on 07/21/06)