Here goes my two cents that I learned from grad school. People who know me closely know how I am sort of all talk and no walk. So, yes, I can't manage them to follow always (I wish I could). But I thought I would share them with you as I feel they are pretty useful.
Learn to read and write Apparently there is a lot you can learn about how to read and write papers. Reading is essential to learn (and do research) and without writing you can't communicate your research with the rest of the world. Try to learn how to read a paper quickly and efficiently without getting lost in the details of the paper. While you write, check and double check whether your write up is comprehensible and grammatically correct (I discovered how bad I am with the use of "the" during past few years). Try to show your write-up to someone who is not a co-author of the paper early enough before the submission deadline. Remember, it is never about sounding smart. A lot of the times people feel they need to sound smart in the write-up and come up with something that is vague and hard-to-read. Also, be patient about the write-up. A good writing should take a lot of time and revisions. So, don't give up, rather try to learn from your mistakes.
Be independent Don't always wait for your adviser. He/she is most likely to be very busy. So you need to be proactive and take charge of your research. Keep an open mind to what others are doing. Go to the events and talks that take place in your department. Share your research ideas and listen to others.
Learn to take care of yourself The sooner you learn to cook the better. I don't know about other countries, but in the US it is kind of an essential skill to lead a healthy life. Dining out is expensive and unhealthy (the healthy options are much more expensive, specially for a grad student). Also, learn to take care of your body and mind. Most of our works demand us to be stuck at our desk. That negatively impacts our health and productivity. Find some form of exercise that works for you (i.e., something you can do regularly). Grad school is a long haul, long enough to not ignore your health.
Be positive This is probably the best thing you can do (not just in grad school, but in any where). I know it is a cliché, but still I had to include it in the list. No matter how greener the grass looks on the other side, life throws lemon at the people living there too! Being happy is a precondition to be successful. A lot of times, when things get harder (or does not go your way), you will push yourself beyond your boundary. So, stay positive in your work and in your words.
Don't procrastinate (too much) This is something I want to learn. So please let me know if you find something on this :)