If you start at Murray Street Mall and head west down Murray Street (past the intersection with William St), you will come across Tiger Tiger (great cafe), El Dorado (clothes), Miin (local clothes designer) and the Belgian Beer Cafe, which marks the intersection with King Street.
From Shafto Lane, head back to William Street and cross the train tracks to get to Northbridge. This site has a map with all the places to visit and the links of individual businesses. The street is full of little independent clothes and art retailers and has managed to remain free from chain stores. The local LGBT press "Out in Perth" provides these handy guides to food and shopping in Northbridge (recommended even if you're straight like me).
Right next to the Perth Train Station, visit the Art Gallery of WA, which also has an excellent store. There are weekend markets next to the Art Gallery that are admittedly a bit lame. It would be remiss of me not to mention the PICA, which has a fine gallery.
Also around the city centre is King's Park, which is probably Perth's top tourist attraction (and deservedly so) with the best views of the city skyline and river, plus the park, an Aboriginal art gallery, a design store (Aspects), places to eat, etc. Having said that, I live down the street and I probably haven't been there for over a year (must take up jogging).