Everyone is checking out the state library, why don't you pay a visit to the Melbourne city library? It opens at 8am Monday to Friday if you are an early bird!
The closest library to here in Flinders Lane, a great space to relax and read, plus a cafe and lending service.
If you wander down the other side of the city and need free wifi this is a great spot
“One of Melbourne's most iconic places to indulge in coffee, cake, gelato & now lunches - delicious! a very popular destination. One of my favourite places! ”
“Very close! Reasonably priced drinks, lots of seating options (outdoor, rooftop, main dining room and cool little separate bar that's cosy and has it's own little bar with service).”
“Fabulous little french delicatessen, with a great array of pates, cheeses and other delights. ”
“Not as busy as the other cafes - but I love the coffee here. Also free WIFI and a cocktail bar at night.”