If I plan to stay around the 6ieme arrondissement, which airport is easier to fly into? Is shuttle service available or is taxi the way to go? Is there a direct train into that area of the city?

Orly is a much more user-friendly airport than is CDG. There is bus service into Paris from Orly, and you could use the RER. If you take a taxi, the distance is about half of that from CDG.

Where is your hotel? RER/Metro will get you anywhere.

http://www.ratp.fr/plan-interactif/

Both have RER links, both have Air France bus.

http://www.lescarsairfrance.com/en/our-network.html

If you really have a choice, Orly is closer than CDG to the Left Bank, so I'd choose Orly if I really had a choice.

You should check the www.ratp.fr website for details on the RER/Orlyval thing as it isn't as simple as just a direct train or see this http://www.orlyval.com/

Depending where you are going, you can also use the Air France bus or Orlybus.

None of the airport buses gets you very close to the 6th.

The RER is very close but in spite of ORY being nearer than CDG, the RER to ORY costs 10.90€ vs the 9.25€ to CDG and the ORY route requires a train change at Antony.

A taxi to ORY will cost about half of the 50€ average price to CDG. However, as has been pointed out, ORY is much friendlier airport for the traveller than is CDG.