email forwarding questions.

I have an email address with my university but it has a small capacity. I made an address with gmail and I'd like to use it to forward messages from my university mail. I don't want it to say that the author of every message that I forward is me because then it becomes more difficult to figure out what the email is for. Is there a better way to do this so that I can keep my messages on a place that isn't my computer? Heck, ideally I'd like to just log into gmail and send messages like they were from me at the university address but that probably won't happen. I just want to make the gmail address almost invisible and only used as a storage place.
Thanks for your help!

 

Okay thanks for all of the input, I am able to use gmail to both receive and send messages appearing like it is from my university account. I was curious if there is any way to import all my messages to my gmail account while preserving who they are from (i don't want them all to say they are from me). Is there a way to do that?