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

Always get the best recording you can find. If you get a bad recording, chances are you will not listen to it. So it's a waste. Sadly the more revealing the system, the worst bad recording sound.

Yes, get the best recording you can find for the price you want to pay. 

If you can't move the speakers, or change the room, that is 2 huge things that are keeping you from getting the "best" from your system. If this is because of your other half, try plants, rugs, or canvas wall hangings. In my room, all the pictures are deep canvas, this allowed me to put sound insulation in the backs of them. Not as good as the real thing, but way better over nothing. 

If you are stuck there, the only real thing left is equipment, try some cables, power cords, maybe a better cartridge? Then isolation devices, spike your speakers? 

You didn't say if you have a power conditioner, that is good easy thing to get, as well as audiophile power outlets. There are good ones that will not break the bank, and do make a difference. 

Personally I do not like tuning the system with power cords and interconnects. It's a total crap shoot, with a high price of entry. On my system, PC made a difference, but not sure it was worth the cost. 

For me it’s all about listening to the music and not being too analytical, and I’m an engineer.  But tweaking is a fun part of our hobby.

I love the sound of tubes but I don’t trust myself or hype to pick them for my setup.  I have always gone to Brent Jesse at audiotubes.com for his recommendations.  It’s amazing the experience he’s had with a tremendous variety of equipment manufacturers.  He gets lots of customers feedback on the success of his recommendations.  What’s interesting is the best sound in my equipment does not come from “names”.  And the preamp tubes have a much bigger impact than the power tubes, but you already know that.

Just my experience, but I don’t know shit

For me it’s all about listening to the music and not being too analytical, and I’m an engineer. But tweaking is a fun part of our hobby.

I love the sound of tubes but I don’t trust myself or hype to pick them for my setup. I have always gone to Brent Jesse at audiotubes.com for his recommendations. It’s amazing the experience he’s had with a tremendous variety of equipment manufacturers. He gets lots of customers feedback on the success of his recommendations. What’s interesting is the best sound in my equipment does not come from “names”. And the preamp tubes have a much bigger impact than the power tubes, but you already know that.

Just my experience, but I don’t know crap

Agreed about preamp tubes, but driver and power tubes can make a big difference too.

I have to say for me it's the power source. The are some excellent advice here today.