There's plenty of answers discussing the problems with having a cable hanging into the Earth's atmosphere - the drag, the heat, the resistence on the orbiting craft, etc. However, I don't see much consideration of the orbital mechanics of the scenario, which would be an issue even if there were no atmosphere.
Simply put, once a craft is in orbit, all that keeps it there is its speed. If you had a craft that could somehow catch the bottom end of the cable at ground level (as it zips by at around twenty-something times the speed of sound), then you have a massive difference in speeds to resolve. If your cable was somehow strong enough (and/or stretchy enough) to bring the lower craft up to speed without breaking the cable, and the contents of each craft were somehow protected from accelerations that would turn cargo and crew into paste, you're now about to run into conservation of momentum. As the cable speeds you up, you're slowing the upper craft down... which means that it will stop orbiting and fall back down.
Even if you put your upper craft in geostationary orbit* (and lengthen the cable considerably to still reach the ground), as you climb up the cable, you're still pulling the upper craft down. Precisely how this works out depends on your relative masses; if you're about the same mass then you'd meet somewhere in the middle. Theoretically you could then push the upper craft down, perhaps by extending a massive pushrod, to continue on your way to orbit, at which point you've invented a rudimentary space elevator. Indeed, some space elevator concepts involve having something very massive (like a captured asteroid) in geostationary orbit*, with a huge tether to the ground - the huge mass means huge momentum and inertia, which mean that its position isn't affected as much by things climbing up to it; an effect that can be further minimised by sending something back down to balance out each thing that comes up.
* Actually, it would have to be a little above geostationary, so that when the mass of the cable is taken into account, the centre of mass is at geostationary altitude. There's also challenges to overcome with regards to the tendency of the whole system to want to rotate relative to the Earth, so that the tether breaks away and drifts out into space, but that's tangential to the question.