Matt, until you can get well away from Melbourne (100km from the outskirts is what you need), concentrate on the Moon, Saturn, Jupiter, Venus and brighter star clusters (Jewel Box in Crux, Wishing Well cluster in Carina). With a pair of about 10x or 12x binoculars held steady (on a tripod is good) you can see a lot. Search through these threads over the last several years - lots of good information on choosing binoculars for astronomy. And a great article from Dennis:-
http://www.iceinspace.com.au/63-374-0-0-1-0.html
For meteor showers, you really have to be out under dark skies and be patient. And be prepared to be up early. Meteor showers are often best seen in the few hours before dawn. But you will see sporadic meteors all night long - about 4-5 per hour and some can be quite startling bright (perhaps 1 or 2 per night). Here is a good site for meteors and lots of other observing information:-
http://home.mira.net/~reynella/skywatch/ssky.htm
Have fun! Congrats if your girlfriend also enjoys observing!
