you might want to visit www.sba.gov and www.score.org You can find elements there about writing a business plan. You might want to work in a similar type of place for a few years before starting out on your own. There are probably some communities out there that want to start something like this, and you may be able to get all of your funding from this. About 80% of new businesses fail within the first 5 years. That's something to consider. Unless you are getting funding to operate a place even at a loss, it's tough to make any kind of business successful. I would say going the non-profit, community sponsored route might be the best. Don't put any of your own money in. You might be able to set it up in existing community space, like a town or city building, or space donated by a community organization.
Those things sound fun indeed. Fact is though that when I was around 15/16 I was always bored to death in the youth centre in my village. Everything organized for us was lame, the room was lame, the games were lame, it was well ment but just a drag for most. I gues we just wanted to decide for ourselves what to do and most important: no adult supervision. Not that we were just focussed on devious stuff, we just wanted time and space for ourselves and have the feeling no one was deciding it for us.
There was a place like this when I was a kid... I was too busy with other stuff to actually go there but it seemed to be popular. I think it may have had some outside funding from charities and/or the city.
They used to do that at one of the bars about an hour from here like once or twice a week. Open for 16-20 year olds, no booze. People started having sex in the booths and stuff, they closed it down.
The places I mentioned were all strictly business ventures feeding off of the boomer crowds. No religious, charity, self improvement affiliations at all. So they sank or swam depending on pleasing the patrons. There were abuses, such as sneaking in booze, or drinking before entering, they would do "breath checks" at the door. And a fight might break out now and again, but it was a different time. They were some of the social centers for kids, in addition to the local fast food restaurants where you would cruise around "the circuit".