I will use unwaxed dental floss. what is the proper way to floss to not damage my gums but to clean well enough?

Back and forth: through the contact, between the teeth, gently moving gingivally. Once through the contact, pull towards the buccal (your cheek). Do not snap floss between your teeth into the gum tissue... you will impact bacteria and debris into the sulcus.

Very simple: Guide the floss between your teeth using a gentle sawing motion. Gentle is the key word.

Gently guide floss: I would suggest to use a dental floss, such as Glide, as it is easier to get between tight contacts. Gently wrap the floss between the teeth and go all the way to the gingiva, on both sides of the teeth. If the floss is not brought down far enough, the results will not be adequate. Again, remember the key is gently.