You're allowed to take a salary working at a non-profit, so long as it's an arm's length transaction (e.g. based on average salaries for those duties at similar organizations). The organization is educational, so its purpose is an exempt one under section 501(c)(3).
Raising angel/VC money is next to impossible as a non-profit, so your primary sources of funds would need to be public donations, philanthropic grants from private foundations, or grants from local/state/federal governments.
Non-profit status will prevent you from lobbying, and will require that a substantial amount of your business income comes from the exempt purpose (i.e. tutoring). Any income not from that purpose is called unrelated business income, and is subject to income tax (this is called the UBIT).
There are also no shareholders - if the business really takes off, you are still only entitled to your salary, not proceeds from the business. When the business is dissolved, these proceeds are distributed either to another 501(c)(3) of your choice or by a court.
With the exception of the difficulty finding funding, these restrictions probably aren't that onerous to your organization. Consider how you'd like to fundraise (e.g. by soliciting donations?) and weigh these limitations against the exemption from tax on income related to the charitable purpose.
Also note that recognition of 501(c)(3) exemption is a fair bit of paperwork and can take about 6 months.