Hello everybody,
I'm French and live in Lyon. I would like to be able to advise you on the best area but it really depends on what you are looking for. I've been living in the Bellecour area so far and really loved it: it's really the center of the city, very easy to get everywhere, lots of shops and restaurants.
the 6th arrondissement is more posh (also more expensive I guess)
If you are after something more typical you should go to Saint Jean (=vieux Lyon) or Croix Rousse. But it's almost impossible to park a car there.
The place you really should avoid is Guillotière because it's not safe.
If you are looking for a house with a garden then it's best to go a bit outside the city center: Ecully, St Didier aux monts d'or, Villeurbanne etc....
In January I will be moving to Tassin la Demi Lune which is also outside Lyon (20-30 min from the center by public transport, 15 buy car). I'm looking for a flatmate and I would be very happy to share a flat whit an english speaking person so please don't hesitate to contact me and ask me questions. It's a ground floor flat and there is a very cute garden.

Actually if I can be of any help for anything, just ask!