Make a Hollywood resume. In Hollywood, no one cares that you work at the mall, but they do want to know that you have been on a set. The easiest way is to start with student films. They always need help. If you have no experience, then you will generally start at the bottom as a Production Assistant. This job is where everyone screams at you, demands ridiculous stuff and generally pay you nothing. But if you survive and do well, they will call you when another project comes along. One rule is not to get stuck in one position too long or you will never go up the ladder. The rule is no more than 3 movies at a certain level.
Write a regular resume and cover letter: If it is a major studio, most of the work will be done online. Sell yourself as with any other job.
Make a list of movie studios you want to consider. Disney has a great program, but it is very competitive, and there are smaller and more accessible studio internship programs out there (see Resources below).
Be enrolled in college or graduate school. In general, companies give college credit because it is cheaper than paying someone. Some studios do offer a stipend, but those are rare.
Be available at least 3 days or 20 hours per week. If it is during the summer, you should not have a problem. Otherwise, you may have to juggle a hectic schedule of classes and (mostly) unpaid work.
Be prepared. If you show up not ready to work or like someone who has no clue, it will not go over well. Ask questions when necessary, but never, ever disturb others with questions that can wait. If you are lucky enough to get on a movie set, time is money.
Work hard. If you are one of those people who likes taking shortcuts or doing the least amount of work possible, you're looking into the wrong business. People refer people they've worked with and liked. They even bring them on new jobs with them. If you are a complainer, always late and have oodles of simple questions, look for a different field to enter into.