I suspect you're looking at a more casual, informal situation but you still should look at running your club like you would a business organization (minus the profit implications, of course). So you'll need a president, secretary, treasurer. You'll want to check into your state's laws on setting up a non-profit club or even the legality of running an informal club. You'll need to get a strategy in place (essentially a business plan) that covers advertising, scheduling, insurance (doubly check out liability insurance), supplies (ribbons, numbers, clipboards, etc.), bank account, potential locations (written contracts here are a good thing), etc. You'll want to get as many volunteers as you can - be kind to your volunteers as they can burn out easily if overworked. Check with your volunteers to see what kind of expertise they have that will be useful to your club. Plan meetings with your club that, in addition to working on upcoming events, give your members the opportunity to voice concerns and ideas.
Work wise I think you`ll see considerable effort to get it started then effort should taper off when all the responsibilities are assigned and everyone gets into the swing of things. Be prepared, though, that at some point the club may flounder a bit when the novelty wears off and competitors and volunteers drift away and then you`ll have to do another push to get things back on track.
As loveduffy mentioned check with your state horse council. Where I am we have a provincial one - it`s a good place to do some networking and gathering of information.
Best of luck with your endeavour. You`ll have to post pictures of your first show.