Often during the interview process, you will be given some guidance of the dress code for the position or company. There is an understanding that your are in school and may not have an extensive work wardrobe, but if it is a conservative, formal environment, you will need to invest in some basic dark pants/dress shirts and ties. No need to spend a fortune either, consignment shops or resale locations are a great place to find professional wear and nice ties at very cheap prices. This is true for both young men and women who need office wear while also in school.
If the work environment is business casual or very casual, the first day or week, make sure to still dress as nice as possible. Clothes should be free or noticeable or sloppy wrinkles, fit well and start out with simple basic coloring. For women, nothing garish, no crazy shoes or tons of jewelry. NO FLIP FLOPS! Once you work in the environment, you will get a feel for what everyone wears, who is considered to be appropriate and who might not be, and you can take your queues from there how to dress the remainder of the time. But I caution to never emulate someone who is a trendsetter or pushes the limits on the dress code. Remember, you are an intern and the impression you are making is important and could lead to a possible future position. Internships also can come with questionnaires and feedback to your school and department and how you dressed should never be a topic unless it's positive.
If your internship is outside an office environment for much of the time, there are usually guidelines for what to wear or even a uniform or clothing provided ( polo shirts with logos, etc.). If you will be on your feet a lot, I heartily recommend if you spend on anything, make it a comfortable pair of appropriate shoes.