Skylark4's points are spot on.
There was a thread on this a while ago. There are some items that I'd say are mandatory - like "Will I get on with the other members of this group? Do I want them flying "my" aeroplane?"
Are the financials "right"? I've seen groups that were really "flying clubs" where the owner made good money and the others subsidised him. If all members aren't totally equal, I'd walk away.
Is the aircraft the type you want to fly? If you're a local area bimbler, don't buy into a long-range tourer or an aerobatic mount just because that's all there is in your area.
Some parts of the world have lots of groups, and shares in the "right" aircraft are easy to come by. Others are very sparse.
DON'T try to form your own group without doing your homework thoroughly - there are many pitfalls!
But if all works well, you get to fly a nicer aeroplane that anything in the local clubs, and for a lot less.
Going rates vary depending on type, but I'm in a group of six on an Arrow III, £100 a month each, £65 an hour wet, and occasional top-ups for nasties like FM immunity and top overhauls.
The more hours you fly, the more you save compared with hiring club aircraft.
There are many more considerations (Private Cat or Public Transport; learning on the group aircraft, group rules...) It goes on for ever!