It depends on how things run at your school, obviously, but I've been doing some looking into starting a club myself. A couple of us are trying to get a lab animal club going at ISU (student chapter of ASLAP), hopefully getting it off the ground first thing this fall.
Stuff I've come across:
Here at Iowa State, clubs are considered official "student organizations" which requires a little hoop-jumping (not a ton, but have to set up some paperwork, have "leadership" training and at least 3 officers, etc). That recognition gets us accounting stuff for the treasurers, etc. We also need to have a faculty member/mentor for the group, who signs off on stuff, and come up with a club constitution, to define who and what we are.
If you want to be a student chapter of a vet organization, there's sometimes some hoops to jump there, too - minimum number of student members, national chapter fees, etc. Again, usually not a big deal, and gets you recognition/use of names, but again takes a bit of work.
Around here, getting a new club up is pretty simple - send out an e-mail asking for interest, try to find a little startup funding for the first couple meetings (vendors can be very willing to send you a few hundred bucks if you sell it right), find a faculty sponsor and a few students willing to volunteer to serve as officers and spread the load, and you should be on your way. Our hope is to get a few dozen students who are in it for the free meals, at minimum, and get a few simple wetlabs/lectures going - just animal handling, unique struggles, etc.