In the first place remember that just because the business world has gone casual does not mean that you should. Confusing 'casual' with 'business casual' is a good way to show everyone that you are not keying into the attitude that the company is looking to foster, and it can make you stand out like a sore thumb. The first day you go in, make sure that you dress well and keep an eye on what everyone else is wearing. This can give you a good idea of how it is acceptable to dress for work. It is always better to be overdressed than under.
Another thing that you should remember to do is to ask intelligent questions. While there is nothing wrong with figuring things out on your own, remember this is a new job with new things you are not familiar with; getting the right information at the beginning can go a long way towards getting you the job that you need later. Remember that, intern or not, you are still doing real work for the company and you should take your job seriously.
When it comes to making sure your internship goes well, monitor your behavior at work. While some media consumption on the clock is okay, remember to unplug it if it looks like it is going to be a problem. Remember that formality is something that can go a long way. Unless you are going to be in a place where you are sure that what you are doing is acceptable, then you should stop and reassess your behavior. Make sure you are not emailing informally, if that is inappropriate for your job setting, or in a manner that makes you seem less than respectful.
Internships are difficult to come by, and if you are in a place where you are looking for them, take the time to try to make a personal connection. Cold calls can work in your favor, but don't neglect sending an email to the human resources department. Adding a personal touch to your application is something that more and more interns are learning how to do, and it can teach you a lot when you are looking to stand out.
When you are working your internship remember that you should be as social as you can. If you are shy, this is not going to serve you very well at all. Remember that being intelligent is not going to do you any good if no one knows about it. Get in there, talk to people, and ask them what they do and what they are looking to do in the future. This can help you make contacts that will serve you well in the time to come. This is also an opportunity to have others assess the work you are doing and to get feedback. Never miss out an opportunity ask a good question or to figure out how something works at the company. Having the ability to learn and the desire to learn is something that you need to cultivate.
Consider what your options are when you are looking at getting that internship that you are after. Try and find one that lies in your area of interest so you can gain experience and see if it is something you really want to do.