I was going to try to answer but I don't know a ton about the hunt world. I know that VHSA (Virginia Horse Show Association) and BHSA (Battlefield Horse Show Association) shows are basically anyone can ride in them with or without a membership but to earn points you have to be a member. They have rules on how many shows you have to be in to participate in the Championship shows.
I saw there is a new "league" that looks very interesting called the sterling and marshall league: Marshall & Sterling League That one looks like there may be qualifiers involved.
As for rated shows, someone will be along shortly to help out I am sure!
A rated show is a "AA", "A", "B", or occasionally "C" rated show that is recognized by "USEF"(and "USHJA" for the hunter/jumpers). "AA" & "A" shows are usually Wed.-Sun. With lots of classes to choose from, they're by far the most competitive & is usually a expensive show(my show checkout bill is usually $800-$1,000, sometimes cheaper if I win prize money in classes) for some of the best riders & horses. "B" & "C" shows are closer to a local level, and are usually 1-3 days long, they aren't nearly as competitive as a "AA" or "A" show & are quite cheaper, yet still are recognized by USEF, the points don't count nationally or zone-ly. For all rated shows(atleast in the H/J) you do not have to be a member of USEF or USHJA to show, but it helps because if you are not, you get hit with a non-membership fee(ours are usually $30 per association). USEF keeps track of all of your placings(in both rated & non-rated divisions) but points count for national/zone(a few states in your area) rankings in rated divisions only. When doing classics, or medal classes, at rated shows, there are certain leagues & associations(Marshall & Sterling) you must join to either participate in the class, or to qualify for the rideoff for that class. If you haven't shown before, I would definitely start at a local(not recognized by USEF, but could be recognized by your local or state association)or a "B" or "C" show. (:
Also, most rated shows do not need qualifying for. Zone finals need to be qualify for by doing other "A" & "AA" shows earlier in the year to be top 25 in your division in your zone. Rideoff classes must be qualified for earlier in the year by doing well in the qualifying class(es) earlier in the year.