You can ask the USCF office in Crossville if they can send you a copy of pamphlet on how to start a chess club. Such a pamphlet should come to you if you register as a club affiliate of the USCF. The pamphlet may exist on the website somewhere. If it is, it is not easy to find.
One of the negatives of the Rulebook of Chess, 6th edition, and in fact all of the official Rulebooks, is that there is no information on how to start and run a chess club. That information was in the Official Chess Handbook, and the older Harkness Bluebook, which are hard to find. Because of space limitations, the Rulebooks do not have the club information you are looking for. You would think that little bit of advice would be in there for those ambitious enough to start a club. But no, all the Rulebook is concerned with is tournaments. It is assumed that you know all about clubs or have the books mentioned above. In addition, you would expect to find in some FAQ on this website, “How do I start a chess club?”, and then point you to a pdf with the information. That should be under the Club section on the website. All you will find is a list of clubs affiliates in the states.
Get the library to agree to do it within a specified time range. Some libraries have conference rooms or a utility area that is used for events that will give you enough space. Hopefully, you can get the use of the facility for free since you are bringing kids from the community into the library. A donation to the library wouldn’t hurt.
Next, get the following: 10 or more standard sized sets and boards, a demo board for instruction, some beginners books and books with hundreds of puzzles. I would suggest books by Jeff Coakley, “Winning Chess for Kids” and “Winning Chess Exercises.” You can also get other beginners books. Pandolfini’s “Weapons of Chess” is good for some general lessons for slightly more advanced kids. Get books as need be and give them to the library to create a chess section so that kids can borrow them. Your wife or significant other will be relieved if you give all of your chess books to the library!
When you feel you are ready to run tournaments, buy the Rulebook. Download pairing sheets and wall charts from the USCF website. Make copies of the pairing card from the Rulebook to use, or make your own. You don’t need software to run small events, plus it helps you and the kids to learn how to run a tournament. Use one of your tournaments to raise money for the library. That will help to keep them happy with all of the kids you are bringing in. To USCF rate the tournaments, you can then purchase club affiliate status from the USCF for $40.
Establish a set of rules of behavior for the kids. Impress on them that their behavior determines whether you can keep your library site. A bunch of noisy kids alarms librarians. You need to keep the librarians happy. Have a mom or three help out. They will keep the kids in line and will be very helpful in running your program.
With luck, you can keep total cost of setup of the club to under $200. As time goes on you might buy a few clocks and books. Don’t forget to give the library a donation once in a while, too. After the start, your expenses will depend on how much you want to spend on chess stuff, copies, promotional materials, stationery, and your own time. Good Luck with your endeavor! It can be a lot of fun.