Personally I prefer it starting on Sunday, but also here with Portuguese being the main language it only makes sense to start on Sunday, Monday to Friday are called second-fair, third-fair, fourth-fair, fifth-fair, and sixth-fair. Fair as in either market or festival.
There are 365 days in a year. If we keep the standard 7 day week, that gives us 52 weeks, with a single day left over. That day can be New Years Day, and be outside the standard week. Call it a special day, make it a holiday, whatever. Leap years would have 2 special days. Put the second in the summer to balance the one in the winter.
That means every year is there same. Divide the year to into as many months as you want, as long as each one has a 'multiple of 7' days. January 1? Always a Sunday. April 3? Always Tuesday. Leap year day? It's on the calendar between Saturday and Sunday.
I'd pull a xkcd, we'd have 13 months every year because it's a prime number, but the amount of days per month would be powers of two for the first 7 (also prime!) = 2, 4, 8, 16, 32, 64, 128; after that, apply some weird math with the current year in order to distribute the remaining 110 days among the 6 months but such that they all end with an odd number of days. Every month would have 2 weeks, made of half the month's days + 1.
Are you trying to anger the gods? Below the fixed stars, there are seven great gods and their lights in the Heavens. And each has its appropriate day. Which two gods shall you so tempt their wrath by denying their day?