What do you look for in a flying club?
Availability, good maintenance, reasonable prices, good policies, and airplanes better than you can rent.
Also, "club" can mean a lot of things. Some FBO's call it a "club membership" when you buy block time. Some FBO's require you to buy a membership and pay dues as if they were a club. I don't consider these for-profit types to really be clubs.
Then there's some clubs that have theoretically unlimited numbers of members. Availability may or may not be a problem, depending on how many of them actually fly on a regular basis. You'd be surprised how many folks belong to a club and never fly at all.
Then, there's equity clubs, where you actually own a piece of the club and membership is generally limited to a specific number of people. With one plane, look for no more than 8 members. Two planes, maybe 20. Three or four planes, 12 members a plane and so on. That should give you good availability.
Then there's partnerships where you probably only have one plane but you actually own a piece of the plane.
With any type of club, you'll want to know how to get in (and how much), how to get OUT in case you move away/don't like it/buy a plane/whatever, how much per hour for the planes (wet or dry?), how much per month for dues or whatever your share of the bills is (and how much) for a partnership, policies on where you can land (grass? field length/width? outside the ConUS?), policies on scheduling and how it's done, policies on taking the plane(s) on trip(s) for multiple days (is there an hourly minimum per day?), how squawks are handled, whether they keep GPS databases up to date if applicable, whether the plane(s) are IFR certified and current, whether you'll need to take club checkrides for currency, what instructors you can use (does it have to be a club instructor, or can you use any qualified CFI), what the experience minimums and checkout requirements are for their airplanes, do they have required meetings or other events, do they have social events or group fly-outs, what upgrades are planned for the future, how the club leadership is structured, and if I can make this sentence run any longer.
FWIW, as an example, I belong to an equity club with 30 members, two Archers (one with GX55 IFR GPS), and a C182 (with G430W IFR GPS and S-TEC autopilot). To get in, you have to buy someone's shares who is looking to get out. "Going rate" is generally $1000 but if someone wants to get out bad enough (and stop being responsible for monthly dues), shares are sometimes given away. There's also a $50 stock transfer fee to get in. As I alluded to above, you're responsible for the monthly dues until you get out of the club by selling (or giving away) your shares to a new member. The planes are currently $87 per tach hour on the Archers and $124 per tach hour on the 182, which includes the fuel surcharge (we have a fuel surcharge structure that automatically changes every month to reflect current fuel prices). Dues are $120 a month but your first $40 worth of flying per month is free (we call it the "flying credit") as long as you pay your bill on time. Our members are allowed to land our airplanes at any FAA-recognized airport, including grass strips and private strips. If you want to go outside the ConUS or take the plane on a trip longer than 10 days, you're required to give advance notice to the board of directors so that the proper insurance and maintenance requirements can be taken care of before you leave. Any overnight trip requires one hour per night minimum time on the plane. Scheduling is done on aircraftclubs.com which also has an interactive voice-response phone system. We have a Maintenance Officer who handles all squawks and regular maintenance scheduling. All airplanes are IFR certified and all GPS databases are kept current. Our only club currency requirement above and beyond the FAA's is that if you haven't flown ASEL in 6 months you're required to take an instructor (no going up solo to regain currency after that long). We're allowed to use any qualified CFI. The Archers require that you are at least a student who has soloed, 25 hours, and a CFI sign-off. The 182 requires either 100TT and 5 in type, or 10 in type, and a CFI sign-off. We have a "wash day" twice a year where we wax the planes and make 'em look purty (required), and an annual meeting (optional). Social events are mostly limited to going to the Jet Room for lunch after waxing the planes, unfortunately. We have a board of directors consisting of a President, VP, Secretary, Treasurer, and Maintenance Officer. As far as upgrades, we're hoping to sell one Archer and buy a Diamond Star, and do paint and interior on the other Archer. (The 182 has been upgraded about as much as we're gonna - 2001 interior, 2005 paint, Garmin 430W/340, MAC 1700 digital flip-flop #2 Nav/Com, S-TEC 50 2-axis autopilot... It doesn't get much better. Maybe an engine monitor someday...)
Hope this helps.