Welcome a-board! You've come to the right place for help. It's always a good idea to do some of your own research to back up what you read here, but there's tons of experienced PTOers who can help you out.
Typically, as a PTO matures to the point of becoming a formal, independent non-profit organization, the first step is to adopt bylaws. That's essentially an internal document, but will be submitted to the state and IRS as part of the several steps to becoming "official". Next comes getting a tax id (like a social security number for a group) number from the IRS - free and easy to do at
www.irs.gov, using IRS form SS-4. Next, incorporate as a non-profit organization in your state. Rules vary from state to state, but it's usually easy to do, and low cost. The "charter" you've heard of may be part of your incorporation applicaiton, but it's most likely a type of organizational document. In our state, we don't have a "charter" per se, but we do fill out a form called the Articles of Incorporation in order to be incorporated. It's not as hard as it sounds, and small businesses (that's you!) do it every day.
Next is the hardest step, but not impossible. Apply for recoginition as a federally-registered 501c3 non-profit tax exempt charity through the IRS. It's a big application, with a pretty sizeable (though one-time) fee, but it makes your group as official as any other charity.
You might want to order a copy of the PTOToday Start Up Guide (from this website). It includes a lot of info about becoming official, and has item by item instructions for the 501c3 application. It's a great place for detailed reference.