So what do you do with yourself all summer if you've been accepted to graduate school, but are waiting for the fall to begin your studies? Before I start, I'd like to say: Congratulations! You've worked hard to gain admission to graduate school, and you deserve a pat on the back just for getting this far. That being said, it's not going to be a piece of cake. Perhaps the greatest error of many graduate school students is going into the experience expecting something similar to undergraduate work. This is not the case!
Graduate school requires self-motivation. It is typically centered in research and involves a significant amount of independent work and learning. Professors are not looking to disseminate information so much as they want you to find the information on your own and learn to work independently.
While your first year may be centered heavily in classroom learning and include the exams that you would expect, even your first year is likely to be much more of a "full-time job" experience than your undergraduate work ever was. Therefore, one of the most important things you can do now is prepare mentally for the challenge that you will face. Studying and working with like-minded people who are dedicated to the same field of study as you will be exciting and challenging - but you need to keep the right attitude to enjoy it. You need to know why you are there, doing the work, and keep that in mind as you engage the material you'll be studying.
You may also want to consider working in order to earn extra money. If you were lucky enough to earn an assistantship or grants that will pay for your education completely, good job. However, most people will come out of graduate school with significant loans and most people are too busy during school to do any outside work. Therefore, saving money before you enter graduate school can be an excellent plan.
If you are moving for graduate school, you should use this time to find housing and prepare things you will need during school (whether that be a local bank, local dentist, or whatever.) When you begin graduate school, you don't need the hassle of organizing a whole bunch of details you could take care of now.
While you are in town finding housing, it is probably worthwhile to stop by your school and meet with the graduate advisor. You can talk about your program, what courses you might be taking, etc. Have some idea as to what you will discuss before you arrive, however. The graduate advisor will now be seeing you in the light of a future colleague, someone who will definitely be in the program. Be prepared to discuss classes you've taken in the past, your interest in teaching, or other relevant matters.
Overall, relax before the challenge ahead. You will be working hard, exercising your mind and your fortitude over the years of your graduate school experience. You want to enter graduate school in a calm state of anticipation rather than a frazzled state of anxiety. Just remember why you wanted to go to graduate school in the first place and let that sustain and guide you from the shock of the initial coursework through to the completion of your program.