Hi, and welcome!
I can't say anything about having a baby here - many of the answers may depend on the particular health insurance plan you end up with and whether your wife and child get a supplemental insurance policy.
Yes, most private schools here are quite expensive, but your children are young enough that they are theoretically good candidates to transition smoothly to the local school system. They should have time to learn French before the academics in a local school get very rigorous and before academic streaming decisions are made.
If you do a search here on English Forum - my preference is using Google as in this link
, you'll find many discussions about the pros and cons of going the local school vs. the international school route. Once you are familiar with the basics trade-offs, you'll probably have more questions to ask here about the specifics of your situation. Most schooling questions are posted in the Family matters/health
section of EF.
I also suggest you get a copy of Margaret Oertig's book on the Swiss local schools, Going Local
, as a start. It will give you a good background on local schools in general and on programs for new students who do not know the language of instruction.
If you decide to try the local route, you may want to find an after school French class for your daughters as soon as you can, and try to continue French classes during the summer, so that they get a jump start on the language.
Hope this helps, and best of luck.