To start, doing some prep work ahead of time is a good idea. Ask the school you're interested in about their audition process months in advance if possible. Each school has their own tryout method and knowing ahead of time can help you prepare, feel more confident and perform better.
If you have the option to choose your own piece to perform, select something you genuinely enjoy and have a connection to. Your passion for the music will shine through and impress everyone. On the other hand, it's also important to choose something you can likely perform flawlessly. It would be wise to play an easier piece you know will go smoothly over one that tends to cause stumbling blocks. Make sure to practice as much as you need to beforehand and if possible, record your rehearsals to honestly critique yourself. Along with the music, if you bring your own instrument, ensure it's in the best possible shape. Bring all the spare parts and accessories you could possibly need.
As well, increase your chances of getting into a great school by applying to multiple places. Even if you think you won't be accepted, try anyway. Make sure to apply not only at the 'sure thing' but also any school you would be delighted to attend. You could end up with a wonderful surprise.
It's a good idea to do some research into the instructors at the schools you're applying at. Get familiar with their teaching styles and viewpoints to ensure you're on the same wavelength. You may even want to get in touch with a few different teachers in various schools in advance. You need to get along with the instructors who will be a big part of your education and having as much information beforehand can only help in your decisions.
As the big day approaches, ensure you take care of yourself. Eat well, exercise, and most importantly get lots of sleep. If you are flying or driving long distances to your destination it would be wise to get there the day before to reduce travel exhaustion. Keep in mind there could be unexpected issues on the day of the audition and be mentally prepared and adaptable to change.
If possible, find out ahead of time how the audition will be set up so you're ready for any surprises. Be ready for questions from panel members and the possibility they may ask you to stop before you're done or perform something else. And always know where the bathrooms are located!
Keep in mind, you are a student so don't be too hard on yourself. No one expects professional-level perfection and schools are looking for new recruits they can help to hone their crafts. Just be prepared for curveballs, try to relax, and simply do the very best you can. Afterwards, be sure to thank the panel for their time and keep in touch. A nice thank you note would be appropriate and a show of class.
Best of luck.