You only need a BS/BA from an accrediated institution. There are exceptions but this is for sure. The field is your choice and only matters that you enjoy it and do your best. The better your grades the better you look to the selection panel. Extra cirricular activities help as well.

Career paths include but not limited to:
Aviation (the best in my humble opinion)
Sailor (boat/ship driving community)
Marine/Environmental Pollution and Response
Fishieries
Aids to Navigation
Civil Engineering
Naval Engineering
Financial Management
Public Affairs
Personnel Administration/Human Resources
Medical/Health benefits
Tactical Law Enforcement

I'm sure there are more but realize that you will be a manager for the most part. You will have the ability to jump career fields to an extent but the long you wait the harder it is. I would say that the Coast Guard can offer a lot of what the other services do but on a significantly smaller scale.