Be careful, volunteer for too much and you might end up like me - 5th year at uni and still going (I might be done by the end of my 6th or 7th year at this rate).
But, and this is a big but, all the stuff I've gotten involved with at uni has been very rewarding. I've made a lot of friends, had a lot of fun, helped a lot of people and added a few lines to my CV (and the importance is in that order). Some good intro volunteer stuff are Shack Tutoring (if you're confident in your HSC academin abilities), Hype Smiths (if you don't mind handing out info about Roundhouse events) or getting involved with a club (there are a lot to choose from). There are also one off events which need volunteers, Arts Week, U-Week (now known as Sourcefest), Mooncaske Yum-Cha, etc. The Source is the best place to get started. If you're interested, head down to the blockhouse and go into student development, they're in charge of all the volunteer stuff.
As for me, these are the things I've volunteered for/gotten involved with:
Contact - information and referal service
Shack Tutoring - tutoring local high school students
ComSoc - Commerce students society
EcoComp - the Australian Economics/Business Studies Competition
Yellow Shirts - O Week volunteers
U Week/Sourcefest - like O Week without the tours or club stalls, in session 2
Hype smiths - promoting roundhouse events
Oral Blitz - no longer around, but similar to hype smiths
Revues - med/law/cse revue, student run comedy theatre shows
Foundation Day - UNSW's birthday, big event in session 2
I'm one of those guys who you tend to see "doing everything"