I have french Schengen visa. As far as I remember in Schengen there is regularly no border control and no passport control. The rules of my visa are described: I must stay in the territory of France more than in the territory of any other country. I want travel from Lille (France) to Amsterdam. First question: How will the embassy know the number of days that I spent in Belgium and Netherlands? This is necessary for next visa. Second question: should I register or inform officials about my crossing Belgium and Netherlands borders?
You can document your travels to other Schengen countries by saving tickets and receipts. There is no internal border control, so the authorities will only know how many days you spent there based on the evidence you can present. They don't often take a whole lot of interest in that, however, only occasionally. You do not need to make any formal declaration to any officials in order to cross the internal borders, but you may be asked about your plans when you enter the Schengen area, at which point you should describe your plans. If that happens, it's not likely that your plans will be recorded in detail; the officer is most likely just checking whether your itinerary is consistent with a Schengen visa issued by France (for example, that you are spending most of your time in France).