The idea of a space elevator usually requires an orbital anchor point, in which case lifting yourself up the tether would pull you fully out of the gravitational well.

I've not done the mechanics in a long time (not since college), but I believe the tether itself would be providing all horizontal propulsion. Essentially by riding the tether up, it starts pushing you faster and faster. Since the payload's mass would be small compared to the anchor, the drag on the anchor would be negligible.

There's likely some need for thrust compensation on the anchor over time to counteract the delta V lost to the lifting of payloads, but that would all be part of station-keeping and would be there for tether drag as well.