Your best bet is to spend some time reading through this forum. There's a ton of advice and information on this in the old posts, and articles in the magazine archive. You should also consider ordering the PTOToday Start Up Guide which answers start-up questions in one binder.
It sounds like your PTO is very small (at least for now). You probably don't need to formalize as a federally-registered tax-exempt 501c3 charity (that sounds scary, but it's what many mature PTOs have done). The IRS doesn't expect registration until a non-profit group consistently grosses over $5000 per year. For the time being, until your group grows in financial size and organizational experience, you could start out very simply.
You are right. The first step is to get a tax identification number from the IRS. Free, fast. Do not use anyone's social security number, and do not use the school's tax id number. Get your own. With that number, your group can now open a small business checking account. Grace said you'd need bylaws to open your account, but that has not been my experience. If you haven't already set up an account, shop for a small business checking account with very low or no fees that is located convenient to your school. While it's sometimes nice to be able to view your bank activity on line or have an ATM card, I wouldn't select a bank soley on those features. It's far more important to avoid paying fees for the privilege of depositing large amounts of cash, or a fee for low account balance, for example. Read the fine print and ask the bank clerk specific questions. Banking is very competitive - you won't see the same offerings at every bank.
Assign at least 3 authorized signers to the account (PTO officers) and order your checks with two signature lines. Use the school's address, but your PTO's name.
Grace mentioned a VERY common misconception about non-profit organizations. There is no federal limit on the amount of profit an organziation can earn, nor must a group spend down all its profit each year. A federally-recognized tax-exempt c501c3 harity does not pay income tax, no matter how much money they have in the bank. Yes, a 501c3 group does have to file an annual "tax return" called Form 990EZ, but there are no actual taxes due. You can legally earn, bank, and save profit from year to year, without penalty.
Being 501c3 has its benefits (allows your donors to write off their donations, formalizes your group, qualifies you to participate in certain programs such as grants, exempts your group from federal income tax and possibly state sales tax), but it also comes with responsibilities (good record-keeping, annual tax return filing). It's not appropriate for every PTO, especially a new small one like yours.
I'm not an attorney and I don't work for the IRS, but if I were an officer in your group, I would open our checking account, do our good work, and visit PTOToday often to deepen my understanding of these issues...so that in the future if we grow to the point of registering as a 501c3, we have a foundation of knowledge to draw upon.
You're doing the right thing already - you're asking for advice. Relax, you're on the right track and your students are lucky to have you!