Interesting this one never got an actual answer. As noted in the comments, this is (at least partially) addressed in the short story The Churn.
The short answer is - not easy. If you have enough qualification (and what this might be is not specified, other than that there is an age limit), it could take a decade to get selected for an apprenticeship off Earth:
“Anyway, no one’s taking me for a vocational. Waiting lists for that
are eight, ten years long. By the time I came up, I’d have aged out.”
and later
“You’re shipping out to Luna on the noon launch from Bogotá station,
Mr. Burton. These apprenticeship programs are tough to get into, and
last I heard, they take it mighty poorly if you miss your berth. Might
wind up waiting another decade to get back on the list.”
As a minimum you need to be registered as a citizen and not living off-book:
“Starting from here? I’d never make it. I’m not even a registered
birth.”