Essentially read, read, read!
Also, there should be judging handbooks on the AOS site - there certainly are on the SAOC site. Yes indeed, there is a link to a PDF at the bottom of Judging Handbook
That will give you a fairly good overview of the judging process and so on.
What it won't do is teach you much about orchids, but judicious googling, reading a load of books and electronic resources such as AQ+ and OrchidWiz certainly will help. OrchidWiz is particularly cool when you're trying to learn about hybridising, and also contains tons of growing information.
With regards to books, if you don't already have any, get one of the fairly comprehensive encyclopedias (The New Encyclopedia of Orchids, Flora's Orchids or Botanica's are all good, Botanica's rather handier to cart around with you); this will familiarise you with the diversity of orchids and give you some idea of the growing requirements of various orchids.
Then get some more specialised books
on groups you find personally interesting; I've really been enjoying both Stewart et.al's Angraecoid Orchids - Species from the African Region
and Koopowitz's Tropical Slipper Orchids
. The latter sort of gives you an insight into the kind of info most judges seem to have memorised - the major breeding trends and important hybrid lines and the influence of certain species on hybrids. I imagine such books exist for other groups, but I haven't hunted for them.
If you're not a member of a society, join one, and if necessary drive/hitchhike/walk/crawl to get to their meetings, and especially, shows, and as has been said, volunteer! I do a ~270km round trip once a month to get to my "local" OS, but further involvement is rather precluded by distance! I was asked if I'd be willing to serve on the committee, but driving that distance twice a month seemed excessive (committee meetings are not on the same day as society meetings).
I don't think I'd ever know enough about orchids to become a judge, and I'm too selective in my tastes. I hate most cattleyas (actually "hate" is a bit strong "find overly fussy and showy" is closer), and I find some of the paphs horrific (any of the parvisepalums) for example, whilst others are enchanting. It's a shame you can't specialise in particular groups, but so far, orchid judging expects you to be able to assess the merits (and know about) all orchids.