There are many types of propulsion from electric turbine engines to steam powered catapults once used on aircraft carriers. Which of the many types of propulsion would be the most efficient in getting a vehicle to space via space elevator?
Clicking a few links through your linked question provides the answer here.
David Smitherman of NASA/Marshall's Advanced Projects Office has compiled plans for such an elevator that could turn science fiction into reality. His publication, Space Elevators: An Advanced Earth-Space Infrastructure for the New Millennium, is based on findings from a space infrastructure conference held at the Marshall Space Flight Center last year. The workshop included scientists and engineers from government and industry representing various fields such as structures, space tethers, materials, and Earth/space environments.
Current plans call for a base tower approximately 50 km tall -- the cable would be tethered to the top. To keep the cable structure from tumbling to Earth, it would be attached to a large counterbalance mass beyond geostationary orbit, perhaps an asteroid moved into place for that purpose.
Four to six "elevator tracks" would extend up the sides of the tower and cable structure going to platforms at different levels. These tracks would allow electromagnetic vehicles to travel at speeds reaching thousands of kilometers-per-hour.
So, the answer is "Electricity". Theoretically, you'd have a solar panel farm attached to the base of the elevator providing at least the majority of the power. Since the elevator would be at the equator, there's bound to be plenty of sunlight...
Helium. Gravity is the biggest issue. You can't accelerate all segments to geostationary velocity, therefore, you have to offset gravity with another method. The use of He within various parts of the structure - especially lower portions of the shaft - would add to the overall vector compensating for gravity.