For a while at least all traffic would be one way, up. Atmospheric reentry is much more cost effective for all downward traffic. It's easy to get down, hard to get up. The completed beanstalk would probably cover a football field and have many tracks and could spare one for personell transfer, but cargo would just fall down to it's destination shuttle style, but the starter kit is much smaller(single strands) and have one way traffic for the first few decades at least, mostly the "Spiders" that drag the mono molecular strands from the ground up the cable to the end.
I don't want to be picky here but I think that something far more ambitious is possible. But I also think that this evolution into space has to be accompanied by a balancing development back here on Earth.
The fact is, most people on the planet are never going to go into space. Just like most people never leave their own country, never mind migrate round the world like my family did. Migrating, for us, turned out to be a good thing to do because the weather in Oz is much better than UK, but the same is not likely to be said of space weather.
If we do things right however, everyone on the planet stands to gain from the increased knowledge and the potential for peaceful unity that a broad and ambitious global project could bring.
As far as I can see we need a project to increase the standard of living of everybody on the planet by about the local equivalent of US$2000, as that $2000 was valued in the year 2000. This CAN be done sustainably [see my 4th axiom for reassurance] and this will give a far more secure economic base for the size of project that is entailed in this vision for a space elevator.
So maybe a single tube or ribbon like thing may be what we start with but I'm thinking that, further on, a series of continuous belts will be able to provide up and down traffic at a couple of hundred kilometres per hour.
Once a tubular structure is established the spiders keep up the job, making it more and more redundant and more capable of absorbing damage long enough for the spiders to repair it, and capable of lifting heavier and heavier loads and using mag lev technology to launch payloads out to the outer solar system.
I'm thinking that tubular and tapered cables will provide fail-safe suspension for the rest of the system which will involve a series of continuous belts, which obviously can't be tapered, with each to be looped over a big wheel at its top end. Presumably all but the bottom one could be looped around the next wheel down. The ribbon/belts shouldn't get tangled because the Coriolis effect would keep them well apart.
Power for the transport would come from the very large geostationary satellite, nuclear powered probably [see the Adams Atomic Engine web site for a candidate].
Manufacture of all the big parts would probably need to happen on [or IN] the moon, but manufacture of the vast number of small parts, and pieces of machine tools etc, may need to be done on Earth. Getting many metal parts up to geostationary or Lunar orbit can be accomplished by accelerating them up a vacuum tube using electromagnetism/linear induction.
Such a launch tube, at least 20 km long, aimed up at about 45 deg, would need to be at the equator, and would be suspended within/supported by a dome/cone/pyramid shaped bubble filled with de-oxygenated air at whatever pressure was needed.
It's all possible!