Google does have a service for downloading all your google data
Once the archive is created, you’ll get an email to let you know it’s
ready. Depending on the amount of information in your account, this
process could take a few minutes or several hours, but most people get
their link the same day they request it.
If that doesn't work with the University system, you may have to use Outlook or Thunderbird to download that data.
Regarding alerting everybody you have corresponded with, that is a little more challenging. You will have to look though all your incoming email to see who you need to notify.
Many colleges keep your account active for a year after your last class. After that time they either close the account, or make that address available to a new student if they have the same name.
Upon graduation or when you leave the institution, you need to make an effort to modify those mailing lists, or tell friends you email is changing. As you do this, the email arriving in your account should slow to a trickle. For people that need to find you your presence on sites such a Facebook or LinkedIn should make it easy to find you and reestablish contact.
Also don't include your university email address on your resume, because you hate to lose a job because you don't have access to that account. Always include an email address you control, and will control.