All public health insurances cover the full cost of the course, yes, it is free, (if you are private, please consult with your insurance to see what they do & don't cover) and you don't need to contact them first. All you need to do is find the course you want to go on, ask if and when they have spaces free and join. In your first class you will likely be expected to bring your health insurance card with you and you will sort out the small amount of paperwork, including likely signing in advance for each class as there will possibly be times you can't come due to illness, conflicting appointments etc). The person running the course will then deal with the payment through the health insurance company so you don't need to worry about any of that.
There used to be a time limit on starting the course before your baby turns 3 months, but this no longer applies. If you are going to bring your baby with you, it would be inpractical if your baby is already on the move so make sure to do the course before then.
You are welcome to bring your baby(s) with you to the classes and a little area will be set up for them in the middle of the room, often with a few toys and mats/blankets to lay on. The classes are very relaxed towards a mums needs and you needn't worry about having to feed your baby mid-class or your baby crying. Bring comfy clothing such as jogging trousers and a T Shirt or vest and you can wear either socks or go barefoot.
There aren't many classes available in English as far as I can find. Fokus Leben,Hebammenpraxis Corry Finné and The Center Berlin have previously offered English language Rückbildungsgymnastik, so it is worth calling/emailing them to see if they will have any classes in the near future. However, you could also search for what classes are on near you and find out if the teacher can speak English.
If you don't mind doing the class in German, then you can simply follow what the other mums are doing. I speak good German, but don't understand all the terms used, so often I just look at what my neighbour is doing.
Post-baby you can also take a number of classes, paid for by your health insurance, called Rückbildunggymnastik. You can read more about this here.
For more information on Germany's health system, read here.
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