What organization to join

Look at all your options.  In the case of soccer, there are a number of national organizations, including US Youth Soccer and AYSO (American Youth Soccer) with which your new club could become affiliated.

Once you have decided on the national organization to join (if any), identify the persons to contact at the state, county or regional level to find out how to become a member and enter teams. You will probably end up speaking with the registrar at the regional/county level who will tell you what you need to do, most likely including completing a variety of insurance and other forms and then appearing at a meeting of the league's Board of Directors to obtain approval to enter teams.   Be prepared at the board meeting for representatives from the existing club to argue against the admission of your club to the league.

If the league in which you want to play has a rule against more than one club from a town being a member, don't let it be a road block: just see if you can get your club under the umbrella of another national organization.  In the meantime you may need to find teams your kids can play against if the existing organization will not let you join.  During the first season you may need to resort to playing scrimmage games against the teams in the existing organization.  This can actually be a fun way to keep the children playing for the season without being overly concerned about league standings.

Board of Directors

In order to affiliate, many national organizations will require that your group create a non-profit corporation headed by a board of directors.  At the very least, they will want to know who is running the club (president), who is handling the money (treasurer) and who is keeping the records (clerk), and will require that you provide a driver's license and social security number for each.  Of the three, the treasurer is the most critical person to select first. A person with a financial background (ideally, an accountant) willing to be treasurer can make things a lot less confusing.

Money matters

Money will need to be collected from parents to cover such costs as registration fees, officials fees, uniforms, and field permits.  It is best to identify a person in the group who will serve as the treasurer early on, as handling the money can be rather daunting job.  With no money in the bank and without a large pool of players you may need to ask for help from a local business in exchange for printing their name on the uniforms.

The treasurer will need to set up a bank account and the board will need to declare the type of organization you are to become.  Even if you incorporate as a non-profit corporation it will take time before you can obtain 501(c)(3) status from the Internal Revenue Service so you can accept tax deductible contributions.