There is no public transport as such, so for getting around, just go out on the street and flag somebody down and tell them where you are going. If they are not heading your way, just wait a minute or 2 and try again.
Car hire is not possible, unless you fancy hiring privately – which could be a bit dodgy considering insurance. Your accomodation should be able to organize a hired taxi for a day at a reasonable price.
Between the islands there are 2 options :
By boat will look like this
This boat leaves from near Chindini which is about 1 hour or so drive south of Moroni. The trip across to Moheli is supposed to take approx. 2 hours, however some travellers are saying it can take a whole lot longer if the weather is a bit rough (which it often is). There are no fixed schedule as such but in general : Take shared bus from Moroni around 7 am, arrive Chindini 8.30 am depart around 9 am and arrive Moheli at around 11 am – that is, if all goes to plan.
Alternative is by plane which looks like this
This is operated by AB Aviation and turned out to be pretty good (I was on this plane myself). Its a 20 minute flight to Moheli and the plane looked surprisingly good and very well maintained. Oddly, you have to be at the airport 3 hours prior to departure, so overall traveltime, flying vs. by boat, is pretty much the same (if starting point is somewhere around Moroni.
Cost for taxi / boat hovers around EUR 100-130 return, and airfare around EUR 110, also return. Flights though will have to be booked well in advance as they are often full.