Bonnie - there's lots of info that address your questions already posted here. If you use the search feature over to the left of the screen, you can find tons of info. Grab a cup of coffee and have fun!
In short, PTO's are usually non-profit organizations, so basically, NO, they don't pay income tax. However, if a PTO is a federally-registered 501c3 organization, then it is required to file an annual "tax" return to the IRS, but still does pay any tax $. The decision to apply for 501c3 is based on the PTO's income level, and it's "organizational maturity" which means is your group ready to take on the additional responsibilties of being a 501c3 from now to forever? Basically, if your annual income level is usually over $5,000, the IRS expects your PTO to apply for 501c3, otherwise you could be considered a business, and thus subject to federal (and state?) income tax. I'm not sure the IRS has ever chased down a non-registered "big" PTO, but I suppose it could in theory.
Rules vary from state to state, so there's no one answer about state income tax. It seems that most registered non-profits are exempt for income tax and sometimes state sales tax, but you'd want to do a little research for your specific state.
It's good financial practice, whether or not your PTO is a 501c3, to have your financial records reviewed every year by an independent party. The reviewer could be a committee of your members (but no one who has authority to sign PTO checks), a volunteer accountant, or even just a parent who has a head for numbers and is willing to do the work. Some PTO's pay a fee to have their books audited, but many handle it with volunteers.
This probably sounds confusing and I'm notorious for giving way tooooo long answers here, but it's not that bad. There's a wealth of education on these questions available here, and you can get more from the PTOToday start up guide that you can order from this site, too.