The plans I've seen are that the elevator cars climb up the cable. They need some power source from above or below for this. Generally lasers or masers are considered. The cable is anchored at the top and the bottom. The bottom is anchored to the Earth, any large chunk of bedrock will do. The top has some weight out past geosynchronous orbit. So counterweight isn't holding the cable up. The cable is holding the counterweight down.
Climbing up the cable then is a matter of crawling up from the ground. Transit times I've seen vary from six to eight hours to a couple of weeks. It really depends on how fast the climber can climb. It's climbing a bit over thirty five thousand kilometers after all.
If you don't go that far, you're not at an orbital velocity.
Time for this is around three weeks if you are climbing at 60 KPH. It's around 17 hours if your climber can climb at around 2000 KPH. Wheeled vehicles have been used for most tests. They can make the former speed. Magnetic levitated vehicles can make the latter.
Given the climbing times, most plans have multiple cables called for. There will be up cables and down cables. The cable has to be able to support both itself and the total weight of cargo and transport on it, plus whatever tension is created by the counterweight.
It's an interesting engineering problem.
We don't have a good material yet for making the cable. The best so far is some plastic coated carbon nanotube rope. I haven't seen any yet in lengths longer than a few centimeters. The Japanese are working on it. They claim to be able to make lengths of up to a kilometer. That may be marketing speaking however.
But the lengths needed are more like fifty thousand kilometers per cable however. That's why nobody has tried yet to actually build one. We simply can't do it yet. It's getting closer to possible however.
Kevlar would work fine for the Moon or perhaps Mars. Steel is possible, but the steel cables for the Earth would be literally thousands of kilometers thick at the top. Not really doable.
There are other problems as well. Those include dealing with meteorites, debris collision in orbit, vibration induced by atmospheric winds, power transfer, and flammability issues.All of these seem solvable, but it's not an easy problem.