A doctoral program is a heck of an opportunity. But useful experiences aren't just given to you, you have to go out of your way. Most graduate students realize that. Most gradaute students are also given plenty of good advice about which opportunities to pursue, most of which I agree with. It's great to publish, great to get a lot of training in the really specific thing your lab does, great to get some random hobbies like pottery-making. But I think there are three things to do in graduate school that people just don't tell you.

Learn Broadly

Ironically, I think what some graduate students don't realize (and I don't think I realized) is that the most valuable thing to concentrate on getting out of graduate school is education. Obviously, figuring out exactly what your lab does and start doing it as soon as possible is the first priority. But if you're in a decent program, you are being paid to learn and that's an amazing thing that you shouldn't forget to take advantage of.

After the first year or two, the day-to-day flow of graduate school emphasizes getting things done (doing stuff for your lab and trying to publish). Getting one publication in exactly what your lab does is a very valuable learning experience and helps you build a productive research relationship with your adviser and perhaps other researchers. But my opinion is that over-prioritizing publishing can hurt your career.

What should you do instead? Take more classes, go to workshops and conferences, read broadly and do other things that have no immediate value for increasing your pub count. Optimally, you should do these things in a slightly different discipline from your own. If you care about your home discipline and the other discipline, you'll find a way to combine them eventually. And if you don't, you will still have learned important things. I can't emphasize enough that the key to the value of these activities is that you learn about things you think are interesting. As long as you do that, your future work will benefit from a broad perspective more than from specific publications and just maybe, you will grow as a person.

Care about Teaching

You will almost certainly have to be a teaching assistant at some point, and will possibly teach your own class.

People rarely tell you to care about teaching, but it's a good opportunity. If you're in graduate school, you're most likely considering going academic. But even if you don't, you will likely be doing a research-related job once you graduate. One of the biggest differences between doing research in academia and outside of it is that in academia, you're most likely also gonna teach. In fact, it's probably going to be a big part of your job. You should expend effort to try to explore teaching to see if it's a good fit for you.

In addition, and perhaps more importantly, it's a place in your work where you can see the impact if you do a good job.

Make at least a friend or two outside of the university

This one probably won't advance your career (though you never know), but just do it if you get the chance. It's good for your sanity. Trust me.