1. Ask questions. In high school, classwork came really, really easy to me (you can hate me now). In college, I quickly learned things would be different, but I also had this notion I should keep my ignorance to myself. This notion followed me into grad school and is dumb, dumb, dumb!
Your professors want you to come to office hours to discuss the course material (assuming you are doing all the work and keeping up in class). They especially want you to dig deeper into the material and have fun with it, and then come ask questions about your digging. Going to seminars with fancy, famous speakers is great, but only if you get something out of it! If you are confused, ask. I guarantee you are not the only one. I was one of those people who was always afraid to look stupid, and I missed out on my opportunity to ask the experts questions about their science. Now I have to learn that stuff myself, without the benefit of an expert.
2. Attend seminars (and pay attention). You never know what you will end up doing. It is really important to keep up in areas other than your research sub-field. I was so sick of my project after my PhD that I completely changed sub-fields for my postdoc. I was able to pick something else I was excited by from the great seminars I saw as a student, which narrowed down my interests to a manageable degree.
3. Push to attend meetings. In my PhD program, students could attend one meeting on the department. This was not advertised, and my PhD advisor only really encouraged his Golden Boy (tm) to attend meetings with him. Well, fuck him--I want to go too! I asked around a little, and lo and behold there was a fund to help students with tightfisted advisors travel.
When I found out about this money (it is a dollar amount per student), I was pretty annoyed, but I also wanted to get the most out of it, since advisor wasn't going to give me anything else. By proper budgeting, I used my money to attend 1) local society meeting, 2) International technique meeting which happened to be in my city (I volunteered to help out to defray costs), 3) national society meeting in a far away city (also did some volunteer work there), and 4) local society meeting in a nearby city. If there truly is no travel money in your department, apply for grad travel awards.
Meetings are super important for networking, job hunting, inspiration and feedback. You need to go if you want to stay in science!
4. Decide what you want to do with your degree. I was one of those idealistic idiots who just wanted to get paid to do research without any real plan for what happens next other than "industrial research, I think." This post show what a huge role dumb luck played in my career. This is not a good idea. You should think about what you want to do next so you can tailor your PhD to get the experience you need to actually reach that goal. Do you want to teach? You might want to do some more advanced TAing. Do you want to work in industry? You should meet with all the speakers from industry and try to work on an industrial collaboration if possible. Work at a government lab? You need to work on getting contacts. Whatever your goal is, there are things you can do to help yourself.
5. Let your advisor help you. It is what they are there for, and not just in the lab. You need to communicate when you are having problems. Your PI isn't a mind-reader. An example--I was assigned to TA a very time consuming class that I hated TAing. I was assigned it 3 semesters in a row. The 4th semester, they assigned it to me again (over my objections). I complained to my PI and he said "I didn't know you hated it so much! You should have said something earlier." He got on the phone. An hour later, I had a much better TA assignment, and I never was assigned my hated class again (yes, I TAed a lot--see point 6).
6. Be aware of your PIs career trajectory. My PI was winding down. He took a lot of students in a short period, but soon after stopped taking new ones. And he stopped writing new proposals too. So this meant we were cash strapped. If I looked carefully, I would have seen that he was getting towards the end of his active research career, but I fell in love with my project and didn't care. Projects come and go, but your funding comes from your PI. If they stop writing, you start TAing (in departments where that is possible).
7. You are at a major University--take advantage! Even though you are in grad school, you can probably see great theater, attend wonderful music performances and enjoy great art. Take advantage of the other things found at PhD U. You definitely have time to do stuff outside the lab, so go for it. Your PhD is a marathon, not a race. You don't want to burn out early. I actually took an intro class in another department for fun my second to last year, though this option is not available everywhere. I definitely started paying more attention to the art museum after 3 years at PhD U. Why did it take me so long?
8. Pick your advisor, not a project. See #6. I don't regret my choice, but I also followed an a-typical career path for a TT prof. Your labmates are also crucial, so make sure you meet your potential group before you join.
9. At most Universities, you can find someone to deliver pizza to any building at any time. And nothing tastes better at 2 am than hot, fresh pizza with your labmates!
10. Have fun! I really enjoyed my first 4 years of grad school. I loved learning, I loved my project, I had hobbies, I had a life. I could even save some money (I bought a few shares of Apple stock at its historic low. If I had more money to burn then, I would be rich now!) The hours could be grueling, but I was also working almost completely independently and following my interests. My labmates were smart and fun. If it is all painful, you should quit. Life is too short to suffer for some letters after your name. If you hate everything about grad school, you should reconsider a career in science.