Hi Kevin,
avoid Scandinavia if you want to stay cheap, and prefer Portugal, Spain, Croatia, Greece, Italy, Hungaria, Tzchech Republic.
October is low season for southern Europe, but the wheather is still warm.
My suggestion is to do a tour in Spain/Portugal (Lisbon, Algarve, Sevilla, Cordoba, Toledo, Madrid, Barcelona) for ten days on a rented car, then from Barcelona to Genua with the ferry (a cabin for 4 is 250 euro one way), then with another rented car or using the train (surely cheaper but maybe not relaxing with 2 kids) 10 days in Italy (Genua, Cinqueterre, Pisa, Florence, Rome, Naples, Bari), from there in Greece, another week... But remember that usually the fare will be expensive flying from Australia to two different destinations, and the same for the car.
Another option is to fly on Milan, rent a car and from there do a circuit so:
Turin (it's my city, and it's worth the travel! October is time of vintage on the hills...)
southern France (vintage time): Avignon, Provence, Nice
Italy (Genua, Cinqueterre, Milan, Venice)
Slovenia & Croatia (Istria region, Dalmatia Island, Split)
take the ferry from Split (Croatia) to Ancona (Italy) and go on to southern Italy through the east coast (Gargano Peninsula, Bari, Lecce, Salento peninsula) and back on the west coast (Amalfi, Naples, Rome, Siena, Florence, Bologna) until Milan.
Ask me, I'll be happy to help you with some idea.