You don't say how long you will be in Thailand for. I think this could alter any advice - less than 2 weeks and you'll need to hurry a bit if you want to go woofiing. 1 month, then more time to enjoy the travelling.
The easiest is to fly on a connecting ticket from France (just change planes at Bkk airport). This is also the most expensive.
Next easiest is to transfer to low cost airport (Don Muang) on airport bus and fly on an airline like AirAsia to Chiang Rai.
Both of these mean you see little of the country or people. I would ask why bother going to Thailand.
I would suggest going into Bangkok for a day or two. Then get a sleeper train to Chiang Mai. Maybe stay there a day or two, then take bus to Chiang Rai.
As said, plenty of accommodation in Chiang Rai. If in doubt, you can book something on sites like booking.com. also, make sure you are able to carry your luggage otherwise it will be taxis and tuk tuks for getting anywhere.
Enjoy.
