The answers so far seem pretty good, but I'd like to try a slightly different angle.
Before I get to atomic orbitals, what does it mean for an electron to "be" somewhere? Suppose I look at an electron, and see where it is (suppose I have a very sophisticated/sensitive/precise microscope). This sounds straightforward, but what did I do when I 'looked' at the electron? I must have observed some photon that had just interacted with that electron. If I want to get an idea of the motion of the electron (no just its instantaneous momentum, but its position as a function of time), I need to observe it for a period of time. This is a problem, though, because I can only observe the electron every time it interacts with a photon that I can observe. It's actually impossible for me to observe the electron continuously, I can only get snapshots of its position.
So what does the electron do between observations? I don't think anyone can answer that question. All we can say is that at one time the electron was observed at point A, and at a later time it was observed at point B. It got from A to B... somehow. This leads to a different way of thinking about where an electron (or other particle) is.
If I know some of the properties of the electron, I can predict that I'm more likely to observe an electron in some locations than in others. Atomic orbitals are a great example of this. An orbital is described by 4 quantum numbers, which I'll call $n$, $l$, $m$, $s$ (there are several notations; I think this one is reasonably common). $n$ is a description of how much energy the electron has, $l$ describes its total angular momentum, $m$ carries some information about the orientation of its angular momentum and $s$ characterizes its spin (spin is a whole topic on its own, for now let's just say that it's a property that the electron has). If I know these 4 properties of an electron that is bound to an atom, then I can predict where I am most likely to observe the electron. For some combinations of $(n,l,m,s)$ the distribution is simple (e.g. spherically symmetric), but often it can be quite complicated (with lobes or rings where I'm more likely to find the electron). There's always a chance I could observe the electron ANYWHERE, but it's MUCH MORE LIKELY that I'll find it in some particular region. This is usually called the probability distribution for the position of the electron. Illustrations like these are misleading because they draw a hard edge on the probability distribution; what's actually shown is the region where the electron will be found some high percentage of the time.
So the answer to how an electron "jumps" between orbitals is actually the same as how it moves around within a single orbital; it just "does". The difference is that to change orbitals, some property of the electron (one of the ones described by $(n,l,m,s)$) has to change. This is always accompanied by emission or absorption of a photon (even a spin flip involves a (very low energy) photon).
Another way of thinking about this is that the electron doesn't have a precise position but instead occupies all space, and observations of the electron position are just manifestations of the more fundamental "wave function" whose properties dictate, amongst other things, the probability distribution for observations of position.
In hydrogen:
- It incorrectly predicts the number of states with given energy. This number can be seen through Zeeman splitting. In particular, it doesn't have the right angular momentum quantum numbers for each energy levels. Most obvious is the ground state, with has $\ell=0$ in Schrodinger's theory but $\ell=1$ in Bohr's theory.
- It doesn't hold well under perturbation theory. In particular, because of angular momentum degeneracies, the spin-orbit interaction is incorrect.
- It predicts a single "radius" for the electron rather than a probability density for the position of the electron.
What it does do well:
a. Correct energy spectrum for hydrogen (although completely wrong even for helium). In particular, one deduces the right value of the Rydberg constant.
b. The Bohr radii for various energy levels turn out to be the most probable values predicted by the Schrodinger solutions.
c. Also does a lot of chemistry stuff quite well (as suggested in the original question) but I'm not a chemist so can't praise the model for that.
Best Answer
First let's consider a different situation. Light waves have polarization. If you imagine a light wave coming out of this screen, its electric field can be polarized vertically, horizontally, diagonally, etc., and this is also true for each individual photon.
If I pass a photon through a vertical polarizing filter, I only ever get two results: either the whole photon gets through or nothing gets through at all. So I'll only get two possible results from the measurement: vertical (gets through) or horizontal (gets blocked).
There exist materials that can rotate polarization. So you might ask, when I put a horizontally polarized photon through such a material, what is the moment when it turns from horizontal to vertical? There has to be an instantaneous jump, because it can only be horizontal or vertical, right? But that's not right at all. The polarization just smoothly rotates, through a superposition of horizontal and vertical, as we can see using diagonal polarizing filters. Just because a particular measuring device can only see two options doesn't mean only two options exist.
The same goes for your question. Now it doesn't really make sense to talk about the 'speed' of a jump because the electrons don't even have definite positions; you're just having one delocalized cloud turn into another. But the orbitals do have definite angular momentum, so you could ask how fast the angular momentum jumps. Same answer as for polarization; it just interpolates through a superposition, even though a measurement at any intermediate point will always give an integer angular momentum.
Perhaps something closer to what you want would be an electron in a double well. Starting in one well, the electron can tunnel to the other. The process is governed by the Schrodinger equation and is perfectly continuous in time. I have a feeling you're looking for a way to travel faster than light and you can in this model, but only because we're doing nonrelativistic quantum mechanics. In a relativistic theory everything would properly obey causality.