This is key to your success in any field or any position, especially your internship. If the company wants to heap coffee fetching and those kinds of jobs on you at first then let them. Some people may feel like you need to earn your dues and their respect during your internship. You need to accept it all with grace. That is a very important lesson that too many people have learned the hard way in their internships. Ask for more work when you have time for it and make it clear that you are always up for anything, especially working overtime.
Be a team player.
One of the hardest things to do when you are starting a new internship is to be both a team player and a person who is able to work on their own. You need to find the balance between being self reliant and still working perfectly with others. This may take some time to do but you need to be working on it at all times. That is the key to your career success.
Internships are one of the most powerful ways for you to get the experience that you need in your chosen field. These are getting to be more and more common as companies and employees alike learn how great they can be.