Yes, you should receive a set of emails - one stating the return was sent and one when the return is accepted by the IRS and the state (if applicable). The IRS generally accepts or rejects fairly quickly (within a day or so) - sometimes the states can take longer, up to a few days.
It's not unusual for the email to get filtered by a spam or junk folder, however - so you can also check your efile status using the TurboTax tool below -