Sorry for insisting, but the whole matter is not totally clear to me. I'm again trying to describe the progression:
1. I put both artifacts into play. I pass to my opponent.
2. I wait for my next turn, but instead of starting it I skip it via untapping Time Vault. I cant to anything in this situation (eg. tapping Time Vault und copy the effect, so that I get the next turn !?? )
3. The next turn belonges to my oppenent. At the earliest in his upkeep I can activate my Time Vault and copy the effect with the rings. My oppenent proceeds with his turn.
4. I have 3 turns coming up: My "normal turn", 1 turn from Time Vault and 1 from its copy effect.
To put it: after bringing the artifacts into play I have to survive 2 more of my opponents turns bevor I am able to go infinite. Is this correct?
I'm asking this because people pointed out, that I have to wait only one turn but I dont know how it is possible.
Your progression is correct. It is impossible to untap Time Vault and then "use it" because you do not have priority to play instants or abilities at that time.
Here is how the exact part works:
1) It is your opponent's end step. They play any abilities they choose, or none if they choose, and pass the turn. Note this is the last time anybody can play abilities before Time Vault untaps (with very few exceptions on cards that read "at the beginning of the next clean up step").
2) They go through the clean-up step. They discard if appropriate, all damage is removed from permanents, and all "until end of turn" effects wear off. No player may play abilities or spells.
3) Play now proceeds to your untap, but the replacement effect of Time Vault skips your turn. You do not actually ever enter the untap step. No player may play abilities or spells.
4) Your opponent begins his turn, and untaps during the untap step. No player may play abilities or spells.
5) Your opponent enters the upkeep step. All appropriate triggers go on the stack, and then your opponent receives priority. This is the first time since the previous end step that anybody has been able to play abilities or spells.
In effect, your opponent will receive 2 turns in a row before you can proceed with the combo.
Untapping Time Vault should trigger Rings of Brighthearth .... Someone might want to skip a second turn

No, Rings of Brighthearth only triggers when an activated ability is played.
"If you would begin your turn while Time Vault is tapped, you may skip that turn instead. If you do, untap Time Vault."
That is a replacement effect, not an activated ability. It does not trigger Rings of Brighthearth.