We could build a Space Elevator on the Moon with current materials such as Kevlar and M5.
http://en.wikipedia.org/wiki/Lunar_space_elevatorLifting many tons of carbon will be very expensive. A stronger version of glass fiber (silicon) that can be manufactured from regolith would be nice.
Mars's moons get in the way of building a space elevator on Mars. However attaching a long tether to Deimos would be nearly as good. A say 5 mile flight to the surface could be used. Alternatively the tether could be winched up and down. Rotovators have also be suggested.
The Earth Space Elevator needs a very strong material inventing. Carbon nanotubes have sufficient theoretical strength but materials currently on sale are an order of magnitude too weak. The coming power beaming challenge may reveal a stronger material.