First, I made my PhD in communications engineering and finished about a year ago.

When I first met my supervisor and told him that I would like to do my PhD, he surprised me by telling me what "doing a PhD is really all about": development of social skill. It's by far not about the technology or the topic you dig into, it is about how you dig into it:

1. Read papers
2. Understand papers
3. Discuss with experts
4. Come up with little ideas first, bigger ideas later
5. Trial and error until you can proof your idea valid .. or not
6. Present your ideas at conferences

Read papers

Well, reading is something that may people know how to do. Figuring out what to read that brings you forward is something else.
When doing a PhD, one might think that reading the common papers of your subject are going to help you figure things out, well, then you are wrong for many reasons:

• Papers are usually written in a way that makes them difficult to access (what sucks big time!)
• Some Papers make assumptions that you don't know about or that may not even be true
• Some papers are just bullshit

Instead, you should read books about the fundamentals .. at most. Go down to the very basics, question EVERYTHING.
Then you will see that many assumptions made are .. weird and you may be able to come up with better ones revolutionizing your field from the bottom up.

Understand papers

As mentioned above many papers are written in a way that makes them VERY difficult to access. In fact, your chances of being accepted at a conference grows if you present your subject in a way
that is not too easy to grasp quickly.
Still, some papers are worth a read (in my matters, less than 10%) because they are well written, describe the subject precisely and some even leave room or hints for further research.

Discuss with experts

Once you came up with a great idea the questions you need to answer to other experts is: How does it work and Why does it work.
During the process of convincing people of your idea you will meat different reactions. Some feel excited, some others feel attacked because they tried to figure it out themselves already and couldn't.
What you can learn from this process is how to deal with people, at least you can learn how to deal with smart people.

However, you can still try to learn how to discuss your topic with people that are not experts in your fields, for instance, your family and friends. Try to tell them what you are trying to solve and what ideas you have had.

Independent of whom you are talking to, the biggest lesson to learn here so: Different people approach the same problem differently and THAT is something that can be very helpful to understand, in general, not just for your topic!

Come up with little ideas first, bigger ideas later

For me, coming up with a new idea wasn't much of a problem. I knew how people have dealt with given problems in history, and I had a new problem to solve, so I took the existing approach and tried to fit it into the new problem. Worked almost right away. However, when learning what was going on under the hood, I was able to come up with an even nicer and more efficient way of doing so. I could only come up with this because of discussions with others that helped me shed light onto the problem from a new angle.

Trial and error until you can proof your idea valid .. or not

Once you knew that there is a better way and have a slight idea of how to proof it, you need to actual work on the proof.
In my case, the idea took me 1-2 weeks to complete, but implementations, optimizations, simulations and proof took me 8 months.
During this time your learn one important lesson: If you fail, stand up and try again. Don't let any failure affect your negatively for a simple reasons:
Even negative outcomes offer something to learn from! Even if it just is: Doesn't work this way.

Present your ideas at conferences

Since I play an instrument for so many years I never had any issue talking in front of people. If you have this issue, then this is already something you can learn from.
The funny thing about conference presentations is: Most people don't understand a thing, because only a couple are in your field and know what you are talking about.
My supervisor once told me, a good presentation consists of 3 parts. First part is an introduction that students can follow. Second part is for those that are in your field
and know at least the problem you are trying to solve, the last part is basically just for yourself, because no one should be able to figure out what you are talking in
real-time :)

Anyway, the lessons I learned from presentations is to extract relevant information from months of work and pack it together so that people can learn something from it.
Another thing is timing (pfff, I was never good at it).

Conclusion

Of course I can only talk for myself but I learned a lot during PhD. Of course also technical stuff, math, signal processing, implementation, yada, yada. But the most important things that I learned or probably social skills.