Will the iPhone notify me with alerts when I receive a new email?
Yes, but only if you have the right Settings configured.
The iPhone will not give you a choice of alert tone, but you can switch on new email alerts by navigating to Settings -> Sounds -> New Mail -> ON.
In order for you to receive new email alerts when a new email arrives in your inbox, you must also navigate to Settings -> Mail, Contacts, Calendars -> Fetch New Data -> Push -> ON. If your email accounts support the feature, new emails will be pushed to your iPhone immediately and an alert will sound.
Alternative settings to save battery life and for accounts that do not support Push include Settings -> Mail, Contacts, Calendars -> Fetch New Data -> Every 15 minutes, Every 30 Minutes, Hourly, Manually. This sets the frequency at which the iPhone will check your email and alert you if new emails arrive.
To set individual email accounts to Fetch or Manual navigate to Settings -> Mail, Contacts, Calendars -> Fetch New Data -> Advanced. Manual will retrieve your email only when the Mail app is opened, which means you will not receive ongoing new mail alerts.