Advice needed: Travelling to and between Paris, London and Berlin

If those are your 3 destinations, then it makes sense to take the Eurostar between London and Paris, and fly in and out of Berlin. The order doesn't matter much. I would choose based on price/schedule for the overall trip and airline/airports(**) for the short flight within Europe. When you look at prices for the short flight, be sure to add in all the extras (there are lots of fees on the budget airlines :-). To save on the Eurostar leg, buy non-refundable tickets when they go on sale, usually 6 months in advance. Budget airline tickets are also cheapest when you book early. Prices just go up as time passes.

(**) Berlin has 2 airports, not sure if one if better than the other. I think they are both easily accessible. London has several, so it depends on where you're headed on arrival. Heathrow has the best connections to central London. Take into consideration how to get to wherever you're going.

Thanks Chani. Appreciate your help! Good to know we are on the right track. :-)

Eddie,
I was in Berlin in June 2015 and flew into Tegel (TXL) airport, it is very close to the city. I took a taxi to my hotel in the Prenzlauer Berg neighborhood and it was very reasonable, 20€, I think. Interesting philosophical conversation with the driver!
Judy B

Berlin's new international airport is years behind schedule and so Tegel remains much the busier of the two old airports. Its only merit is being close to the city centre. Surprisingly few long-haul flights use it as a destination so most of your choices from Australia will involve at least two stops (you might find one-stops through the Emirates or Qatar.) Using Berlin as your gateway in both directions will waste time in returning to Berlin and possibly cost as much as going home from Paris or London. Use the multi-destination search function on many sites to find itineraries that fly into Berlin and out of London or Paris (probably more choice out of London.) Or go in the reverse order if you find a deal you prefer.
Travelling between London and Paris is much more comfortable on the tunnel train. It's just as efficient as flying and competitively priced if you buy the tickets far in advance. Long-distance train travel and short-hop air fares work the same way: Early purchase lands the cheapest fares and there are rarely any last-minute bargains.
For all the train information you will ever need to know: www.seat61.com
For budget air fares: www.skyscanner.com

Look at return fares with Star Alliance carriers - Lufthansa and Singapore Airlines.
Australia to Berlin via Frankfurt/Munich
London to Australia via Frankfurt/Munich

Germanwings, Air France, Air Berlin, easyjet all compete on the route Berlin to Paris so cheap fares are plentiful. Book well ahead.
Eurostar from Paris to London.

London last would probably be slightly more expensive than Paris or Berlin last because London's airplane departure tax on long haul flights can be a nice chunk of change. I did a test fare search for a pair of dates in September and a Sydney-London then Paris-Sydney open jaw ticket was about A$150 less than a Sydney-Paris then London-Sydney open jaw ticket.

BA often has pretty reasonable short haul fares into and out of London if they've reached the time frame where they have to compete with the low cost carriers on price. (And both their hold and carry-on baggage policies are far better than easy jet and the like, IMO)

As an option consider megabus from London to Paris (or opposite). It's an inexpensive alternative.

I generally try to start at one end and end at the other. I'm not sure why you would overfly France then return? I'd probably go London, Paris, Berlin or opposite. In the Spring, I'd go this way; in the Fall, I'd probably reverse it to have more chance of good weather.

If you're just seeing the three cities, a flight is probably a good idea. Berlin is a little out of the way. I'd only suggest a train if you have stops along the way.

I would go fly to Berlin if you get a good fare/route, then fly to Paris and Eurostar to London.

I fly into Tegel (TXL), there is great bus service into the city center every few minutes.