What should a grad student do during summer?
Summer While You Are Enrolled
You should do the same thing you do the rest of the year, but with more focus on research.
Graduate studies are not usually about classes. You will take some classes, to get a breadth of knowledge in your field, and to get up to speed on your specific area of research. But most of graduate studies is reading, creating, and evaluating research. You will read papers, and work on research that you can write papers about.
Summer ends up being an ideal time to do these things, because you're not usually taking or TAing classes, and your supervisor is not usually teaching, so you both have more time to dedicate to research. Particularly if you're looking to have a career in Academia, then having published papers are very important. But even if you're looking to go into industry, the projects you work on will probably be more important to your career than the classes you take.
(This doesn't always apply to Course-based Masters or Professional degrees, like Speech Pathology, Library Science, Medicine, etc.)
Summer Before You Start
Reading your question, I realize this may be what you're asking. If you're starting in September, what should you do leading up to then?
I have a few main suggestions:
Catch up on your personal life. Whether this means taking a vacation, working extra to make some more money, or just relaxing at home, spend some time on the things that you might not have as much time to do during your studies.
In particular, try to get your mental health in order. Graduate studies are usually pretty stressful, and between having to independently motivate yourself, facing deadlines, and reading the work of others, it is very common for graduate students to face anxiety, depresssion, stress, and the imposter syndrome. So if you can get your stress levels to a baseline before you start your studies, it will make it easier to handle the elevated difficulty that comes with graduate studies.
Get a head-start in related skills. The summer before your studies is a great time to teach yourself LaTeX (if you're in a field that uses it), or wor processing, or spreadsheets, or whatever is common in your area. Learn how to use a citation manager like Mendeley or Zotero. Get acquainted with Google Calendar or another schedule managing tool
Get started on your reading. One of the hardest things about graduate studies is that you will spend a lot of time with a lot of very smart people who seem to know a lot. For the most part, they know a lot because they have read lots of papers. So, if you can familiarize your self with the actual research that is going on in your area, the things that you don't learn from textbooks and undergraduate classes. You're not going to be able to do too much just in the summer, but any little bit you do will help the start of your studies be less overwhelming.
It really depends. Some take classes, some take a break, some travel, some continue on their research.
For me, personally, I've spent last summer taking courses to fulfill some degree requirements and I'm doing the same this summer.
Every person is different. Some graduate students are different. There is nothing that you should be doing, but maybe depending on where you are in your studies, there are few things you should prepare. Getting ready for your dissertation? Do some research. Need a break? Take up a hobby or maybe travel a little.
It may be best to talk to your advisor.
You do not mention if you are a doctoral student or master-level student.
Typically, summers should be used to get ahead in your program of study and/or career. Especially if you are not engaged in supported research under an advisor. So, reading papers that will advance your thesis/dissertation and/or writing papers for publication are common summer activities. Planning summer work also helps prepare you for a career in academia, where many contracts are 9-10 months during the year.