Please keep in mind I've been here sice 2001 (so don't cut lettuce anymore, eat gut sausage, gizzards and have tucked into tripes for breakfast) so am a bit out of touch and only visit the UK once or twice a year if that, but I'll have a stab at your questions and hope it helps.
What's the cost of living like?
It's more expensive than the U.K. or at least that's what I think. Food seems much cheaper in dear old Blighty but wine is generally cheaper depending on what you buy. Spirits are about the same prix as they are in the UK. Eating out can be quite reasonable, but then again the bill can sometimes take your breath away too...depending on where you are.
What are the pitfalls?
Medical insurance is best described as expensive, ball park figure about €100 if you want most of the bells and whistles...like Dental and Opticians including...the latter really know how to lay it on probably more than the dentists. A friend of a friend was recently was given a quote for some dental work without insurance and it was a whopping €10,000!!! Luckily most Dentists only want to see you once a year and getting in to see one can be tricky, long wait times for an appointment are possible...same for Opticians. My Mrs made one last week and luckily only has to wait until December 23rd of this year.
Along with Income Tax there's also 'Taxe Foncier' & 'Taxe d' Habitation' which are a guess like council tax so vary on where you live and the services you have on hand. I'm in the sticks so it's not so bad, and they are about €120 per month combined. One is far more than the other but I can't remember which way round it is.
Diesel is about €1,30 per Litre, no idea on petrol...but it's usually more.
The bureaucracy can be as bad as it's reported (they excel at it, nay love it..but not the French people only the Govt. types who love to get their own work wrong at your cost) to be and depends on the region you decide to live in. It can also vary from the commune to commune, how efficiant the Mayor is and is in my experience best left to SWMBO to sort out. As an example, applying for citizenship (thanks to the Brexit caper) required copies of all manner of paperwork...not only our marriage certificates, but parents and also death certs all requiring official registered Frenchie translations at €45.00 per document...about €500 for each of us. They said it would take two years and it did take just that, pluse we did everything on time and by the numbers...I think it was 23 months in total.
Never, ever be in a rush for anything. If you like things doing when you want them doing then you will be annoyed very quickly indeed. As an example, if it says a shop will open at 09,00, it won't. Laid back is laid back, don't be in a rush at the checkout, the customer isn't always right, he's usually wrong and that's not over-egging it lol. Walk into a hardware store, don't be surprised not to be asked if you need any help and the country does grind more or less to a halt through the main holiday season when everyone goes on their jollies.
Areas to look at, areas to avoid?
That's the easy one for me, anywhere far away from other Brits...if anything thing stops integration faster, then it has to be living with other Brits. However that said not everyone wants to do that and it's a far simpler option if you are not wishing to go all out froggy and jump in with both feet.
Learn the lingo, it is essential, no need to land speaking like DeGaulle but it does help and you can pick it up along the way, I couldn't speak more than a couple of words in 2001. Listening to french radio and French telly helped a great deal. The local radio being the best, you pick up stuff without realising. I am now B1 level french, which was one grade above the minimum requirement for citizenship....but no way am I fluent in French, way too complicated for my Infantry brain. The LHSgt.Maj. is C1 which I believe is one below the top level, but she's a girly swat and did it at A Level, she wouldn't say she was fluent either...but she isn't far off from what I can tell.
After all of that, it's a great place to live, out here in the sticks it's laid back, no one bothers you much. The French are pretty nosey generally but do know how to keep themselves to themselves when required. Obviously I am painting with a broad brush there, but that's been our experience. Someone touched on rights of passage and things like access to ponds and fruit trees...yes that can be a problem, but there are ways to deter them.
Someone suggested earlier about taking a long term rent...that's what we did...12 months in two rented places while we looked around for the right place...it's a great piece of advice.
My only real regret is not moving here sooner, it's not all been easy, but I'm pleased we stuck with it. I like isolation and this is the place to find it without going too silly. Also very pleased we went down the citizenship route eventually (cheers Brexit for the push) rather than just being a resident, makes life even easier.