student visa followed by tourist visa

I'm an Australian studying in France on a 6 month student visa which runs out on the 1st of July. After it runs out, do I have another 3 months to travel around the Schengen countries? Do I need to leave the the schengen countries and come back in so that my passport is stamped?

It is still an area where there is no Schengen harmonisation.

It depends on the bilateral agreement that Australia has signed with the Schengen country you intend to visit and the interpretation given to it. Some may request a visa because you have spent already 3 months in the Schengen area, other not. Some may request an exit stamp on you passport.

Therefore you have to ask the Australian embassy, which should know, or the consulate of each Schengen country which you intend to visit.