Do you start fine tuning the system with cables or tubes ?

Assuming you are starting with decent cables, I think you will hear more impact from trying different tubes.  I would start with the input and driver tubes which tend to be cheaper and deliver noticeable changes.  If you can source a variety of vintage tubes, try some of those as most of the commonly used tube types have not been bettered by new production.

The difference between the “free” interconnects and zipchord speaker wire and even modest upgraded stuff is vast, but beyond that, improvements become more subtle.  You will also have to discover what family of wire suits your system and taste before deciding how far up the hierarchy you want to go.  For example, someone who likes the basic sound of Audio Note cabling will not like the sound of Nordost regardless of the price level.

My initial comment would be that the room has to be quiet and controlled enough for this determination. A noise floor above ~35 dB SPL will mask the subtle second-order differences between cables or tubes. If the room isn't reasonably treated, you may be hearing the room when you think you're hearing a tube swap. In other words, the room's contribution is constant, so it becomes perceptually invisible even when it's dominating.

All that said, given a decently controlled room, I think that tubes (especially input and driver stages) will generally move the needle more than incremental cable upgrades past a modest baseline. 

This is a no-brainer. Tubes are going to be more influential because they introduce harmonic distortion, whereas cables do not.

I start with cables, aiming to achieve smoothness and liquidity—not necessarily warmth. In my second rig, which is already on the warm side, I was able to reach that goal using the Veritas Argentum XLR cable between the DAC and integrated amp.

However, in my first rig, which is more bright side of neutral, upgrading cables alone is not enough. While it does improve smoothness, it doesn’t fully get me where I want to be.  I’m now looking into tube buffer or tube preamp options.

It's hard to say without knowing what your base line is.  Both the cables and tubes have to sound good together. 

My experience with inexpensive cables is that they often soften the upper frequencies.  If you bought tubes that worked with them, you might not like those tubes when you upgrade to a detailed and neutral cable. 

I've played with a lot of cables and tubes to find the right mix for my system. 

Tube differences can be easy to discern. It depends a bit on whether you start off with basic OEM tubes, or whether you roll between various third party tubes. Upgrade from basic has been instantly recognizable, in my experience. Third parties may or may not be discernible. Some are also more a question of flavor, and preference may depend on what recording you play. As I listen to a wide variety of music, if any piece of equipment does not make all music sound better, then the it does not matter. I do not keep separate sets of tubes for baroque strings vs. EBM and switch them accordingly. And there is the habituation factor as well.

I have yet to hear any difference in any cable. Yes, I tried, and no, I have not heard any difference ever.