Wow guys/girls!

I did not expect to see so many reactions, I was surely not expecting it. Thank you for the espresso hive-minding.
The problem is still playing, I have switched up the coffee beans, have 3 different bags from which I draw now.
Currently I reside in Turin, close to one of the biggest fresh produce markets in Europe. Coffee is sold at many places and packaged in paper bags when you made your pick. It does not come with fancy plastic bags with roast dates and information, so I have no idea how fresh it is and when it is roasted.
I try to ask for beans for an espresso machine, one would think in Italy(!) one could find decent beans to make espresso. Prices range from 1 euro per 100 gram to 1.75 euro per 100 gram.

The machine still dumps the coffee out way to fast unfortunately.
To answer some questions:
I indeed have a digital scale, accurate to 0.1 gram. I weigh every shot, 16 grams of beans into the grinder. At the finest grind setting at the finest internal grinding setting. (both outside and inside of the grinder are at finest setting)
I then place the scale under the machine with the cup on there and try to shoot for 35-ish grams of espresso out. Usually this comes out in around 13 seconds, which I time with the same scale.

Now the coffee does not taste horrible, but it just does not have the consistency of espresso, its watery and seems not to live up to the potential of the beans. I am really wondering what to do now, at which point do I decide that is not my inexperience but a broken machine? I could practise for 1000 hours but it seems to me this machine is just messed up internally.

I'm no expert on the machine, but it sounds like the first problem is the grinder.

You can adjust the shims (I believe) to allow the grinder to produce finer grinds. If it's 13 seconds, that's not just on the beans being fresh or not.

Edit: The machine itself is very popular. The Breville Barista Express in the United States. The issue is well known.

You should definitely check your grinder adjustment. Check user manual on how to clean and adjust it and let us know

Fendervogel, I'm new to making espresso, too (just started in April). I, too, struggled with learning to pull good shots.

I'm not familiar with that specific machine. My machine, a Breville the Bambino Plus, came with pressurized portafilter baskets. I was able to pull half-way "decent" shots with the pressurized double basket, with coffee ground with our Capresso Infinity burr grinder, but they weren't great.

I acquired a non-pressurized basket and tried some pre-ground Lavazza a neighbor gave me. Even though it was pre-ground and actually a Moka Pot grind, I was pulling tastier shots.

Next I upgraded my grinder to a Baratza Vario.

From there things went downhill for a while. I just couldn't pull good shots. The timing would often be right, but they'd usually end up sour.

What finally got me on-track was comparing my grind to the pre-ground Lavazza and starting over approximating that grind. Then I went to a local roaster and bought a pound of freshly-roasted espresso blend. While I was there I asked them about grinding. They took a small bit of a decaf they had and ran it through their commercial grinder at their espresso setting and let me take that home. I further-refined my grind based on that additional example.

In the past two weeks I'd say nine-out-of-ten of my shots have been good, with several of the shots of that freshly-roasted coffee being to die for

Even the coffee I bought along with my new grinder, with which I'd previously been unable to pull anything but sour shots, is producing good results now. (Now that I've got it right I'm finding it has strong citrus hints--so it wouldn't take going far wrong with it to end up sour.)

Another thing that definitely helped me: Just for grins I made myself a WDT tool. Turns out that thing has been a Godsend.

My process now is to refine the distribution in the PF with the WDT tool, tap the PF several times to settle the grind, remove the dosing funnel, use a distribution tool set very high to improve the distribution, then tamp. (I also upgraded my tamper.)

I'm glad I let the membership here talk me into spending a bit more than I'd planned for the Baratza Vario. I'm finding that just one step either way on the micro adjustment can make the difference between great and ho-hum--and sometimes ho-hum and utter disaster.