How is the acetoxy group ortho/para-directing in electrophilic aromatic substitution?

Why is the $\ce{-OAc}$ group o,p-directing in electrophilic aromatic substitution?

How can I compare the activating effect of $\ce{-OAc}$ with $\ce{-OH}$ and $\ce{O-}$? Is it to do with the number of lone pairs on oxygen?

Only the availability of the lone pairs matters. That has to be discussed with inductive or resonance effects. With an $\ce{-OH}$ group you have a hydrogen atom fighting for some of the electrons (inductive effect), and with a $\ce{-O^-}$ group there is nothing else.
Moving on to the acetoxy group $\ce{-OC(O)CH3}$, the lone pair on oxygen is delocalised into the electron-withdrawing carbonyl group, via resonance. You can think of it as the $\ce{C=O}$ and the benzene ring both fighting over the oxygen lone pair, which makes it less available for donation to the pi system. Consequently, the acetoxy group is still o/p-directing (because of the presence of at least one lone pair), but is not as strongly activating as the hydroxy group (because the lone pair is less available).