10 free things to do in Melbourne (CBD)

1. Federation Square (Fed Square) grab a deck chair and people watch, free wifi.
2. Australian Centre for the Moving Image (ACMI) lower level galleries including "Screen Worlds" (within Fed Square). Also Shaun Tan exhibition until mid Jan 2014. Allow at least 90 minutes.
3. Ian Potter Gallery  Australian art. Allow 1 hour, also in Fed Square. 
4. Yarra River walk from Fed Square, cross the Princes Bridge and walk downstream along Southbank. Allow 1 hour return.
5. Free exhibitions in the Art Centre, St Kilda Road, 3 minutes walk from Princes Bridge - currently Geoffrey Rush exhibition (until Oct 27).
6. Arcades and alleyways (get a map from very helpful Tourist Information booth in Bourke Street Mall)
7. State Library galleries and dome/reading room (world class) Corner Swanston and Lonsdale Streets
8.  Walk up Collins Street to Spring Street, one of the most elegant streetcapes in Southern Hemisphere.
9. At top of Spring Street is the historic Treasury Building, exhibitions, explore underground gold vaults.
10. Parks and gardens - Treasury Gardens, lead on to Fitzroy Gardens; Royal Botanical Gardens on other side of the Yarra. Allow 2 hours for Botanical Gardens (walking from Fed Square).

Most of these are non-touristy, things 'locals' do.