Europe working holiday for an Australian

I'm assuming your partner's British passport is because he has UK citizenship (in addition to Australian).

As an Australian citizen you don't need a visa to enter most European countries for stays up to 90 days per rolling 180 day period, but you will need a residence permit if you stay in, say, the Schengen area beyond that limit. (Which sounds likely to be the case unless you're planning your itinerary specifically such that you're outside Schengen for enough time to satisfy the 90/180 rule -- in which case you'll still need to deal with the rules of those other countries).

An UK youth mobility visa will not in itself count as a residence permit for other European countries.

The good news is that as long as you're accompanying your partner who is exercising his citizen's right of free movement within the EU/EEA, you basically cannot be denied a residence permit. You still need to apply for one, and there will be some processing time, but in most cases it will be fast-tracked, and most of the regulatory barriers to other prospective immigrants will not apply.

In principle the rules apply to cohabiting partners, but it will probably make for smoother trips through the bureaucracy if you get married before you travel. (Nobody says you need an actual big wedding, though).

[With the length of trip you're sketching here it is plausible that you will need to deal with tax authorities in the countries you're staying in, even if no money change hands -- but that is outside the ambit of Travel.SE]

The UK youth mobility visa does not grant you any extra rights outside of the UK. As an Australian citizen, you can however travel to all EU countries for at most 90 days without visa anyway (i.e. even without the UK youth mobility visa). Beware, in the Schengen area (i.e. most of the European Union), it's 90 days per 180-day period in the whole area, not 90 days per country or 90 days per entry. In Croatia, Bulgaria, Romania, and Cyprus, it's 90 days for each country. To make things even more complex, there are some special rules for Australian citizens in Denmark, see Consecutive Schengen area visas from different countries? Finally, Ireland has its own rules but you can also go there without visa.

For work or long-term stay you will need a local authorization or visa from each country. For many countries, one option is to get a “working holiday visa” but in some cases (e.g. France, if I am not mistaken), you need to get it before entering the country so plan carefully. Alternatively, if your partner is a UK citizen and you have some form of registered partnership, you can also easily get a residence card in other EU countries based on his free movements rights within the European Union.

Finally, countries like Russia, the Ukraine, Serbia, Albania, etc. are not part of the EU and have their own rules.