You can leave the space elevators at the top but it is nearly impossible to join them there. In the case of the Earth elevator it is about 1 second every 24 hours. Miss and watch both the Earth and the Moon disappear behind you.
The Earth Elevator's joining point is at GSO (Geostationary Orbit) height. By adjusting the spacecraft's altitude the pilot can chase the elevator permitting a slow docking.
GSO has an altitude of approximately 35,786 km (22,240 miles)
The Earth Elevator climber will have a top speed of about 100 miles per hour straight up.
22240 / 100 = 222.4 hours or 9.26 days. In practice the Earth to GEO trip will take longer because a lot of the trip is at 50 mph (miles per hour).
The joining point for the Lunar (Moon) Elevator is L1 or EML1 (Earth Moon the Lagrange point L1). L1 is 56,000 km up from the surface.
http://en.wikipedia.org/wiki/Lunar_space_elevatorAn L2 Lunar Elevator is possible but because it is on the wrong side of the Moon and being longer i.e. more expensive to build I doubt it will be built first. L1 and L2 are good places to build spacestations.
15 years is too early for manned use of the Lunar Elevator. It will still be too thin ( = tiny payloads) and take several months to climb. The Earth Elevator will be built from carbon nanotubes but sections of the Lunar Space Elevator may be made from fibre glass mined on the Moon.
The time from GEO spacestation to L1 spacestation will depend on the rocket used. An ion thruster rocket can take several months to make the trip, used for heavy cargo like buildings. A chemical rocket takes 2 or 3 days. A sling shot (no fuel) from the end of the elevators also take a couple of days.
Transport between the Moon's surface and L2 spacestation or GEO spacestation is likely to still use chemical rockets just to save time. They may still use hydrogen or carbon propellant brought up from the Earth at very high cost. Alternatively the rocket may use powdered magnesium and liquid oxygen, both of which can be extracted from the Moon.
Moon bases at the north and south poles are unlikely to be connected to the Lunar Elevator but there may be a monorail or cable car connection.
I suspect that somewhere along the line there will be laws banning carbon based and hydrogen based fuels on the Moon, those elements will be reserved for food and hygiene.
Have you seen the new Lunar Rovers NASA is working on?
http://www.nasa.gov/mission_pages/constellation/main/desert_RATS.htmlPower for the spacestations and Lunar Base is likely to come from a mixture of solar power and nuclear energy. At the Lunar equator solar power does not work for 2 weeks out of every month.