Melbourne Airport (Tullamarine) is a about 20km northwest of Melbourne city. Since there aren’t any public trains or trams that travel from the airport to the city, you need to catch a bus or taxi.
We usually use the Skybus service. It is fairly cheap ($26 return per person) and it includes the Skybus direct to
Spencer Street Southern Cross Station and then a smaller shuttle bus to your hotel. The Skybus takes about 20 minutes and then the shuttle bus can take another 30 depending on where you get off.
You should remember to book your return shuttle bus pickup at least a day before departure to guarantee a seat.
It is very easy to get around Melbourne city with the excellent tram network. The city circle is a free tram service that runs during the day. You have to put up with annoying voice over announcements of various tourist attractions, but it is free. For other trams, a daily zone 1 metcard is the best value ($6.50) as you can travel in inner Melbourne all day on any tram (also buses and trains). You can buy metcards in newsagents and convenience stores, as well as from the concierge desk at most hotels.
I misjudged our return flight times last Sunday, so in a frantic dash to the airport we had to catch a cab. It was quicker than the bus (as expected) but it cost us $49 one way including highway tolls. This was a quiet Sunday afternoon so I imagine during the week it would cost a lot more. We got to the airport with less than half an hour before departure but we were still allowed to check in (thanks to Virgin Blue!).