This post was prompted by an email from a friend who is about to start graduate school requesting words of advice (or warning). After I replied to that email, I noticed an excellent post by Prof-like Substance that may also be helpful to newbie grad students, so go read that, too.
The ordering of this list has less to do with importance than the order in which these occurred to me.
The financial stuff (written assuming a graduate program in which the graduate student receives some sort of financial support):
1. Find out the schedule to pay fees for the term (as well as what the prevailing policy is on late payments), and get 'em in. (Even though the part you have to pay as a grad student is likely less than the support you're getting in terms of tuition reimbursement, etc., late fees can snowball.)
2. Find out the schedule for your RA/TA paychecks (assuming you'll have some sort of stipend) and check them religiously to make sure they are neither smaller nor larger than they're supposed to be. Why you do not want to be paid too little is obvious. But, it's also a hassle to be overpaid, because eventually someone who's doing the accounting will discover the error, and you will have to write a check to pay the money back. If your too-big paychecks have gone unnoticed by you except to the extent that they have let you buy fresh vegetables to eat with your ramen noodles, you may not have extra money sitting around when you need to fix the error if it has gone on for awhile.
3. If you're in a situation where you're paid a lump sum at the beginning of the term, find out whether you need to pay estimated taxes (since there often isn't withholding from the lump sum). You do not want to have the IRS on your ass while you're studying for quals.
Integrating into your department and university:
4. Find out which functionary in your department knows how all the gory details of registering for classes, getting an advisor, filing the right paperwork for candidacy, getting paid, etc., work and who is disposed to share this information with new grad students. Cultivate this person's goodwill, regularly.
5. Cultivate grad student friends from outside your department. They will help you figure out which features of life in your department are weird and which are typical of graduate programs in your university. They will also help you maintain something resembling perspective. (Plus, they might know some good, cheap places to eat meals.)
6. Locate the library stacks where dissertations from grad students in your department are shelved. From time to time, browse a thesis or two to absorb the local expectations about format, the appropriate level of detail for literature background and description of materials, methods, and results, etc.
7. Make it a habit to attend the public portion of thesis defenses in your department so you become familiar both with the format of the defense and with the approach of the faculty in your department (collectively and individually) to grilling the candidates. (This may help you develop a short list of faculty you'd be happy to have on your own committee.)
8. When shopping for a research group, spend as much time as you can with the grad student members of your prospective group. Go to group meeting (to see how they interact with the boss and with each other). Arrange to drop in while they're doing research-like activities. Trust your gut about whether this is a social setting that will suit you.
9. Research advisors who already have tenure are often (but not always) more open-minded about the diversity of effective work habits of grad students than are research advisors who are trying to get tenure.
10. Have fun! Grad school may be a means to an end you are pursuing, but it will also eat up at least a few years of your life. Those years ought to be enjoyable as well as productive.