Senior Event Manager at Workday
If you want to focus on a major, check out Marketing or Public Relations. I actually got my start in hotels which is a GREAT place to start. You will learn all about events and it is easy to get promoted. You can also start as a Travel Director with one of the leading agencies (Maritz, EEG). http://www.traveldirectors.com/travel-directors/
While you are in school, check out the major associations surrounding events (CEMA, MPI, NACE). Many have student divisions where you can learn about what they do and meet people in the industry. Ask people how they got their start. You'll find most start out as administrative assistants. You can also reach out to local event planners to see if they need "day of" help as an hourly worker. Finally, if you really want to get in the thick of it, sign up with a temp agency. Most large events need temps as human arrows, registration help, etc. You'll get to see the craziness of large events.
For your question: You can do both. I've worked for companies, agencies (meaning they are hired by the company to do the event planning), and been a freelancer to just assist on events. You'll just need to decide for yourself which you prefer.