Fine tuning the system. Where do you start and how do you proceed ?

Fine tuning for me is not necessarily linear path, always about determining the weakest link. Not being afraid to experiment and fail is required, weakest links may not always be obvious. In the end everything matters, sometimes the smallest changes can make all the difference, tuning is an art rather than science.

Tuning is an art indeed.

Leaving wall current alone - it’s a separate big problem. Leaving the room alone too.

Do you begin by seeking out best LP pressings ? Or tube rolling ? Or cables and power cords comparisons ? Isolation devices perhaps ? Something else ?

How do you go about it ?

I tend to start from the beginning - LP pressings.

I have never felt "wall current" or power issues to be a limiting factor, at least where I reside. Room issues are mitigated by my choice of speakers (Tannoy), "mid-field" setup, and light treatment of 1st reflections. Positioning is key.

I obsessed over tubes for a while and have a ton, but it’s not as big a deal to me now. Choosing the right power tube for the amp can be important.

For cables - speaker cables, specifically its silver content, has had the most effect. I like some good solid silver on this link. Large gauge. It definitely sounds different from copper. Unfortunately, it has become prohibitively expensive in modern cables. I buy used.

The biggest issues are: 1. choosing your transducers, and 2. matching your components and transducers to each other. Call this "synergy" or whatever. This is the biggest factor in getting a system to sound good, for me. At this point, I have collected several of every kind of component. The sonic differences by each component (and combination) are very noticeable, and "portable" even to different systems / rooms. 

For recordings - no I’m not going out of my way to seek special recordings. I have some, and they’re great, but are not available for most material. Perhaps 50 - 70% of my vinyl collection sounds "really good" to "great" as is. And I’m grateful for that. Digital has a lower % than this, but there are some bangers in there too.

Beware chasing details.

I would say if you are a tinkerer, leave yourself room to tinker without commitment.  That is, if you want to rotate 2 or 3 different speakers or a couple of different amps, then do that! Don't chase absolutes.

It's kind of important to know yourself here.  If you are just always going to be playing with your setup then get yourself organized for that rather than a constant stream of buying.

Otherwise, it's important to understand where you want your system to go.  What have you heard that's _better_ and not just _different?  What did that sound like?

Components, speaker wire, amp power cord, interconnects, power cords, detailed positioning, acoustic treatments, vibration control.