Calculus and differential equations will certainly be essential during your physics career. You will take classes in them if you study physics at university, so you certainly don't have to learn it all first, if that's what you're worried about. However, if you want to get a jump start I would suggest the following progression. I doubt you'd get through it all in a summer—indeed, I wouldn't even recommend it since I don't imagine would stick if you did—so just treat it as something into which you can get as far as you like, time permitting:
- Calculus of a single variable. Depending on how comfortable you are with abstract mathematics, you might find it useful to take two passes at this. The first using a fairly informal, intuitive approach and the second with something more rigorous. Any number of introductory texts would work for the first pass. For the second, a book like "Advanced Calculus" by Folland or, if you find all this especially easy, "Calculus" by Spivak. But really, the most important thing at this stage, I think, is to get comfortable with the ideas. You will learn calculus rigorously from the ground up as a physics student, so it will be very helpful to have had a more casual introduction first so you aren't bombarded with strange new concepts.
- Multivariable calculus. Pretty self-explanatory. Generally not needed for first year topics in physics, but will be used extensively in your first full course on electromagnetism.
- Linear algebra. Together with differential equations, linear algebra is the language of quantum mechanics. In addition to be invaluable for quantum physics, an introductory university course in linear algebra is often a student's first exposure to highly abstract mathematics. Going through it on your own could be challenging since it's likely to be the first thing really radically different from any mathematics you've done before. However, attempting may, at the very least, help "prime" your brain for it later on.
- Ordinary differential equations. While most of physics is ultimately formulated in terms of differential equations, you can go a very long way indeed without a formal education in solving them. The types of differential equations you will encounter in your first couple years as a physics undergraduate will generally be very simple. Thus, while a good foundation in ODEs will ultimately be essential, I have it fairly far down the list of summer study material.
- Partial differential equations. Ditto the last point.
- Complex analysis. Very powerful tool, even when you're only interested in calculus of real variables. You should at least know what complex numbers are before your first quantum mechanics class; however, beyond that it won't be needed until senior-level physics.
Those are what one might call "core mathematics" for physicists. There are other areas like group theory, real analysis, and differential geometry that are important in their own ways, but will come later and in no set order. Again, this isn't a challenge to get through everything on the list. Quite the opposite. The most important thing here is that you find books teaching the material at a level you're comfortable with. You can always go back after a casual introduction and learn the material more rigorously on a second (or third, or fourth...) pass. Don't jump straight to the ultra rigorous stuff, even if you will ultimately need to know it. You don't have anyone you need to impress with self-education.
