OK, so we build a suspension bridge or ten first. I can handle that. We’ll still have an elevator eventually. And I think that by “designed with nanofibers”, toadspittle means that they’ll have to be designed with nanofibers in mind: You can’t just take an existing design and replace the steel with carbon. The climbers, though, aren’t too high-tech: It’s a little car with a spool on it. They don’t need to be robotic: A robot has to respond to stimuli in some way, whereas a climber is always going to be doing the same thing.
As for launching from one, it’s not just your first thousand miles an hour that’s free. The further you go out, the more advantage you gain, and if you make your elevator long enough, you can get up to escape speed from the Solar System. Which means you can go as far as you like for free. You’ll still need some rockets, most likely, to change the plane of your orbit (unless your destination happens to be in the plane of the Earth’s equator), but almost all of your energy requirements are taken care of.
There are a variety of related technologies, which would be useful for getting around the Solar System. Space elevators are good for getting anywhere that rotates fast enough: They’d definitely work for Earth and Mars, and I’m pretty sure also for the major outer moons (Io, Europa, Ganeymede, Callisto, Titan, Miranda, Triton), but not for Mercury, Venus, or the Earth’s moon. Anywhere in vacuum, you can use a skyhook, a rotating cable with a counterweight at one end, with the rotation timed so that it’s momentarily at rest when the end touches a planetary surface. You’d lose too much energy if you were dipping into an atmosphere, but they’ll work fine for Mercury, Luna, or the asteroids. That pretty much just leaves Venus and the gas giant planets themselves, but who wants to go there, anyway?
As for power beaming, there have been experiments with unmanned aircraft powered by microwave from the ground. The technology isn’t yet mature, but it’s probably closer than large-scale nanofibers.