Shor's Algorithm speed
The best estimate I know of can be found in Efficient networks for quantum factoring, by David Beckman, Amalavoyal N. Chari, Srikrishna Devabhaktuni, and John Preskill, which gives $72 (\log N)^3$. 

Having said that, a straight comparison of number of steps on a quantum computer versus number of steps on a classical computer is problematic for various reasons. First, as D.W.'s answer says, the number of steps depends on the exact architecture of the quantum computer, which we won't have until one is built. Second, the time required for a single step on a quantum computer is likely to be quite a bit slower than a single step on a classical computer.1 Again, we won't know how much slower until quantum computers exist. 

1 If it was faster, you could use the same architecture to build a classical computer that would be at least as fast, and probably faster because for a classical computer, you don't need to worry about maintaining quantum coherence.