My residence & work permits in Germany are expiring soon, but I want to stay and travel in Europe. What visa should I apply for?

If you have stayed on the residence permit for more than 90 days (or haven't stayed in the Schengen area immediately before starting the permit), you are fine, you can stay for another 90 days visa-free under regular short-stay rules. The time spent under a long-stay visa or residence permit does not count toward the maximum short-stay period.

If you have already made some other short stay or want to stay longer than 90 days, you will need a long-stay visa from Germany or another Schengen country. If you are young enough and haven't use it yet, a “Work Holiday Visa” is probably the easiest to get. You could for example get one in the Netherlands.

Beyond that, long-stay visas are typically hard to get (you need to have a spouse in the country, a very good job offer or meet other stringent requirements). In many cases, it might also be impossible to apply from within the country (which would mean that even if you qualify you might have to return to Australia or some other country where you could become a resident, just to be allowed to apply for the visa).

Finally, there is an arcane way to stay a bit longer without a long-stay visa under the provision of a reciprocal agreement that existed before the Schengen area was created. In your case, as an Australian citizen, it might be possible to stay for another 90 days in Denmark, even after having exhausted the maximum short-stay in the Schengen area (see the “Denmark” entry in the relevant Wikipedia article).