The older you are, and the more value you can add to a company, the better your Internship can be. While you are in high school, the less you probably know about the processes and information required to add value, therefore the company will probably not be able to give you a really robust experience.
I think that the best age to get an internship is starting right after high school, and for every summer while you're in college. Your college career office should be able to give you some good leads on which companies are providing internships - paid or (mostly) unpaid.
One of the hardest things about internships is just identifying internship opportunities, especially paid internships. It's a great way for a student to explore a given career path, but the best internships are the most competitive ones to get - either because of the prestige of interning at that company, or the pay itself. Also remember that larger companies with formal programs start to interview for their interns starting in January or so, and smaller, less formal intern opportunities will start to dribble into the career office as the school-year goes on.
Over the long-run, the most important element is getting that experience, and the name on your resume, therefore if you can get a great internship, even if unpaid, then take it, and learn as much as you can. When I interview candidates for full-time jobs, I like to hear what they may have liked or disliked about certain jobs, and it's totally legitimate to say that you didn't enjoy that internship but you should state what you learned, and how that learning shaped the next career move. e.g. if you took an internship at a law firm thinking that you might want to be a lawyer, then discovered that you hated the work, and now you're interviewing for internships in the financial field because perhaps that was the only part of your former internship that you really enjoyed. Totally legit and shows a level of self-awareness and maturity that is required for these types of internships.
Another way to get information about a given career path is to do some informational interviews. Either ask adults that you know, or go through your teachers or college alumni network to help you get in contact with people who work in that field. You should come prepared for a discussion with a list of questions to ask - look up suggested questions on the internet - when you talk to the professional in your field of interest. Your level of preparation will send the professional a good impression of your skills and maturity, and you could end up on their radar screen for future internships. Also always end the interview with "Is there anyone else that you'd suggest that I speak with?" It's always a good way to help build your network at this age.