Alright, so you don't want to spend the rest of your life doing coffee runs and answering phones? I don't blame you. So where do you begin to make the fastest move up the career ladder? Admittedly, I don't have much experience in this arena, but between some research and what I've witnessed, it seems safe to say it isn't the where, but the what.
Everyone starts on the botton, especially in the music world. That is just the way it is and there isn't usually a way around it, whether its errands for a studio, trainee at a management office, assistant at a label or whatever else your first job will probably seem like you've only moved a few steps beyond interning. Don't worry though, you're not alone, according to this article, a college grad should expect to stay in an entry level position for two or three years.
Instead of trying to expedite the process, learn as much as you can where you are so you'll be ready to move up when the opportunity arises. Build a reputation for great work and reliability and work on building new skills so you'll be more qualified. More importantly, find an entry level job in a field you're passionate about. Assuming that's likely music if you're reading this, there are still many different areas to work in that industry and they all use different skills. Figure out what will make you most happy: live sound, touring, publishing, online, radio, recording, etc. Once you do, that entry level work may not be quite as painful.
My last piece of advice may seem a bit odd since the music business isn't quite the same as Hollywood. However, there are a lot of similarities. So, if you want some more info on how to handle an entry level job in entertainment and advance, check out The Hollywood Assistant's Handbook. It was recommended to me while I was interning in LA and it is pretty informative. It is actually part of what inspired this blog! Not everything in it will apply to the music business, but it is a quick read and has some great advice in general too.