Hi Terry,
there is no "best" way.
Getting there
I would focus firstly on how you want to get to Europe and what your budget for that is. There is no point saying that one website is better than another or one airline is cheaper than another because there are different specials all the time. So work out what you budget is, search around regularly and get the flights that match with what you want.
If you are starting from St Jean the two most obvious European points of entry are Paris and London if you are flying from Australia. You then have the following connection options:
a. Fly to Paris, Train to Bordeaux, Train to Bayonne, Train/Bus to St Jean,
b. Fly to Paris, Fly to Biarritz (next to Bayonne), Train/Bus to St Jean,
c. Fly to London, Fly to Biarritz, Train/Bus to St Jean
There are other options of travelling from London to ports in Bilbao, San Sebastian, Irun, Pamplona by boat or aircraft however will require additional connections and from the past research I did more time and cost (which is fine if you don't mind).
There may be different airline options that travel to Barcelona or Madrid. From there you can easily take a train to Pamplona or San Sebastian and connection from there. But again more time and cost.
From what I can tell from reading a lot of the previous discussions on the same topic, most people fly in to Paris and train to Bayonne.
If you prefer to fly you can go to the Biarritz airport website and see who flies to there. Check here
http://admin.biarritz.aeroport.fr/images/guidehoraire.pdf
Use the list of airlines on that .pdf and go to their websites to work out who is flying at the times and departure points and for the class of travel and price that suits you. You can use websites that search multiple airlines (eg bestflights and skyscanner etc) but these often omit specials that are only available on the airlines' own websites, so I would only use them in addition to going to the airlines own sites. You will likely get the best value for money if you are willing to transit from the main airports (ie Charles De Gaulle - CDG and Heathrow - LHR) to the second tier airports (eg Orly - ORY and Stansted - STN) to get onto the budget airlines like easyjet and ryanair (assuming you want to use them).
Getting home
As far as leaving to get home, you can check around but would likely be just as easy to fly back out via Paris or London. You can train for fly from Santiago (SCQ) to Madrid (MAD) or Barcelona (BCN) and connect easily to Paris or London from there.
You can see which airlines operate out of which airport by clicking on these links:
1. Santiago (SCQ):
http://www.aena-aeropuertos.es/csee.../en/Page/1048690016843//List-of-airlines.html
2. Madrid (MAD):
http://www.aena-aeropuertos.es/csee.../en/Page/1049727006273//List-of-airlines.html
3. Barcelona (BCN):
http://www.aena-aeropuertos.es/csee.../en/Page/1045569607375//List-of-airlines.html
So in summary the market for airline tickets is always changing and you need to be prepared to do some research to work out what travel path you want to take and what is the optimal value for money for you.
Final advice: if you see a good deal get it because they don't last for long. There may be slightly better deals that come up later, but at least you're organised and don't risk paying even more.
Good Luck,
Cheers,
Jason