You can, in theory, start immediately, but the problem is that you start with only one stone.
Each stone gives you a half-hour in Abyssea, no more. There are time extensions
within Abyssea, however, so if you have someone shouting for, say, a La Theine Worms party, if it's going well and you don't get too unlucky, you'll be able to get extensions considerably faster than you use time, and you'll exit with the maximum time on the clock when you decide to go see how hard it is to do limit breaks, lol.
One of the first things I'd work on are the Abyssites (Key Items) that increase the rate at which you acquire stones. They're also retroactive, amusingly enough, so that when you get one, you immediately get all the stones you would have gotten had you had it from the beginning. Or so I'm told.
Stones accrue every 20 hours and
what the fuck am I doing there's a perfectly good explanation of all this crap on the wiki...
GO HERE AND READ.