Some hints:
Always start from the top down - there's no point in getting the floors steam cleaned, and then getting all the crap off the rafters. It'll just get messy again.
Always take as much equipment as you can out of the gym first
Chalk eaters/other sorts of humidifers are excellent - see if you can leave them in the gym for a few days running to get rid of some chalk.
Deep chalk buckets are the BEST, and always teach kids (even rec) that when they rub or clap their hands, to do it in the bottom half of the bowl. That means less chalk flies.
The more often you clean your gym, the better. Ours is vacuumed daily (fortunately, we're attached at a few other professional institutions that have cleaners for us). Wipe down your mats when they get sweaty. Painting your walls is a great way of reducing chalk mess (and you have to sugar soap before you paint).
Basically set aside a week or two once or twice a year, when all the gymnasts are on break, and clean the entire gym. But clean the gym as you go throughout the year (including the pit...once a month is optimal!), and then it makes it so much easier.