Yup, it's actually possible to send and receive my.monash (or any other uni/workplace) email from your beloved Gmail account. Here's how:
Receiving (mail into Gmail)
When someone sends mail to your Monash email address, you can have the Monash mailing system forward it to Gmail. Sign in to Monash Webmail (https://my.monash.edu.au/email/webmail.html) and choose the Options link from the row of links at the top. In the Options page, choose the Forwarding and Delivery link from the left side. There you can specify that email should be forwarded to your Gmail address and then deleted (so it doesn't take up any space in your Monash inbox).
You can compose and send emails from your Gmail account which will appear to be from your Monash address. To set this up, you need to be signed into both Gmail and my.monash. In Gmail, go to the Settings page, then go to the Accounts tab. Click the Add another email address link, and a window pops up asking for the name and address. Enter your my.monash name and email address here, click Next Step, then Send Verification. Keep the small window open because the Monash email site has a bug which makes it show Web page links incorrectly. In other words, you won't be able to verify that you own your Monash address simply by clicking the link in the email that Gmail sent to your Monash address.
Now go to your Monash inbox, open the message from Gmail, and copy and paste the confirmation code in Gmail's verification window. Hit Enter, and Gmail adds your Monash address to the list.
Try It Out
Compose a message in Gmail, and choose your From: address from the drop-down box. The person who gets the message will see the Monash address, and when they reply to it, the message will automatically come to your Gmail inbox. When you reply to this reply, the new message will automatically have your Monash address in the From: field. And so on.