There are two ways of getting to the
State Library of Victoria by public transport. The first way is getting off at Southern Cross Station, walking up to the Spencer Street/Burke Street junction, catching a tram up Burke Street until the tram stops at the junction between Burke Street and Swanston Street, then turning to the left and walking up Swanston Street until you arrive at Melbourne Central Plaza. The State Library is across the road.
The second way is catching a metro train going through the city loop and getting off at Melbourne Central. Personally, I prefer using the second way as it's easier and I don't like waiting for or using trams unless I have to.
However, I think the more important question isn't how to get to the State Library, but why get to the State Library? Unfortunately libraries have earned an unfair reputation as silent, boring places of oppression and solitude and that's simply not the case with the State Library (and the vast majority of municipality libraries).
The Splendid:
There's free activities such as indoor and outdoor chess.

Outdoor Chess

Indoor Chess
On sunny days, bean-bags and boxes of books are made available for reading.

Outdoor Reading
There's free access to computers for printing, copying and browsing the Internet. WiFi Internet for laptops is also available.
It's perfectly fine to talk and arrange study groups in the Information Center (Level 2 Ground) and The Redmond Barry Reading Room (Level 2A).

Information Centre - Level 2 Ground

Redmond Barry Reading Room - Level 2A

Redmond Barry Reading Room - Level 2A
There's plenty of stuff to check out like the Keith Murdoch Gallery (currently featuring the free exhibition of "Rome: Piranesi's vision"), The Cowen Gallery (featuring Australian landscape oil-paintings), experimedia, journals and magazines, newspapers, and genealogy.

Keith Murdoch Gallery - Rome: Piranesi's vision
The Not-So Splendid:
The State Library isn't a lending library so you can't borrow items to take home. However, you can save newspaper articles to USB and take them home that way.
You will need a library card to log into and use the library computers for Internet browsing, copying and printing. However, you can easily apply for a library card at the Information Desk, fill out
an online form, or call them on 03 8664 7004 from Monday to Friday, 10am to 6pm.
-You can't bring bags or back-packs into the library. You have to take what you need out of your bag, rent a small locker for $2 and store your stuff in there
-Personally, the ominous white marble busts in the Cowen Gallery creep me out a little. Then there's the fact that there's only one female bust available and the artist went to the extra trouble of adding nipples. I find this questionable.

Is it just me or is it cold in here?
My favourite place in the State Library is the La Trobe Reading Room.

The La Trobe Reading Room - Level 3

The La Trobe Reading Room - Level 3

The La Trobe Reading Room - Level 3

The La Trobe Reading Room - Level 3
It's a quiet reading room and unlike downstairs, I can often find space to set up my laptop and focus. It's an amazing place, so peaceful I've frequently found people asleep at the private study desks. There's also a section of brass plaques dedicated to those who contributed towards the building of the glass dome. One of the plaques on display has "I miss Bob" engraved into the metal. I find it intriguing.
So please visit the State Library of Victoria and your local municipal library, because while books are amazing and invaluable in themselves, there's so much more to discover.