Do I need upgraded rings for the achievement?
There is an achievement which can be completed by collecting all rings.
Does this mean that I also need to collect the upgraded versions (+1, +2) of the rings which can only be found in NG+?
Or is it enough to collect the basic versions of the rings?
Solution 1:
Yes, you need all the rings versions to get this achievement.
That means, you can't get it until NG++.
From Achievements:
Master of Rings Acquire all 107 rings. (NG++ Required)
Solution 2:
Yes you have to collect every ring including their upgraded counterparts as stated on the Dark Souls 3 rings wiki page:
To obtain the Trophy/Achievement for collecting every ring, you must collect all of the different +X variations available, making a total of 107 rings.