I'd recommend getting allergy cards printed in French, listing the foods you're avoiding. Although, as there's no French for dairy, you'll need to be able to say you can eat beef (assuming not vegetarian).
We found France to be really easy, and have similar allergies. Have fun
Actually if you want to say dairy free you can say "sans produits laitiers" although people are not very allergy aware and may not necessarily realise that this includes butter, yoghurt and cheese.
The health food or organic section of supermarkets is good, but you won't find foods specifically labelled dairy free as you do in the UK - it's more a question of checking ingredients each time. There is a range called Allergo or Valpiform (seems to vary) which does biscuits and cakes and cracker type things that are free from a number of allergens including dairy. The Bjorg organic range also has quite a few dairy free products although you do need to check each one.
Don't know if you are soy free as well, but soy milk is readily available, and I think rice milk too (we get ours from organic shop as DS1 doesn't like the supermarket ones, so I can't quite remember!). Lots of soy yoghurts too - look for Sojasun or the supermarkets sometimes have their own brand of soy yoghurt. I've never seen soy cheese though.
St Hubert Bio margarine is vegan - it's quite hard to find margarine without milk solids in it.
Ham often has dairy in too but the Fleury Michon brand seems to be fine, and most Herta ones are ok too.
Is there anything else specific you will be looking for?
I seem to remember Dad took bread with him - He is Gluten free. bagettes are dairy free buy I don't know if you are completely avoiding that yourself? They don't seem to be very into cereals so you might be better to take your own to be sure you have some. We just said in restaurants that he was allergic - easier. Also DS is allergic to eggs,nuts, fish and sesame. My French is basic but I was able to explain everywhere what we needed to make sure we did not eat. The staff in restaurants were great always happy to check with the kitchen and make suggestions for us. Again we cooked a lot and it is a case of looking at similar things to that you would normally buy and checking the ingredients. Make sure you know the French for lactose as well as milk and any other varients that you need to look up.
Alison is right - baguettes will be fine as they are not allowed to put anything other than a few basic ingredients (not including dairy). Other bread you might need to check (although basic wholemeal should be ok, but not things like corn bread). Supermarket bread there a few brands that are ok but I don't buy them so don't know what they are off the top of my head.
Lactose is the same word in French, also look out for lactoserum and babeurre.
No idea about cereals as my kids just eat bread, though I suspect you might be best to bring your own.
Weta summarised it all pretty well - I'm French myself living back in France now. Oat milk can be found in Carrefour supermarkets (organic section), but if you're after Oatly or other calcium enriched types you'll have to go to an organic store. I'm buying an Italian brand called "the Bridge" (strange name, but really great product). Maybe check beforehand where to find the stores (I can help). I've never seen Oatibix here.
Also beware of cured meats, a LOT of them have lactose.
Just a thought...it may have just been the area we were in, but we searched many, many supermarkets and couldn't find a margarine which didn't contain soya. Perhaps someone else can advise if that isn't the case elsewhere.
Thank you all for your help. I am much better prepared now. Erring on the safe side for the baby and taking the basics for me and will be extra happy if we find a supermarket close by that sells some of the nicer things but will go without if they don't.