Great question! I generally agree with Amy's answer, with a few additional thoughts. I have personally worked in the nonprofit sector for around 8 years with a bachelor's degree. I work as a grant writer although I have no education at all related to fundraising or grants - all my skills come from experience.
I completely agree with Amy that you should first work in the nonprofit sector to gain some practical experience and figure out where you want to focus. Then you can explore degrees such as a Masters of Public Policy, Masters of Public Administration, MBA, etc. - whatever fits well with your area of interest.
In my experience, some jobs will pay more if you have a master's degree but will still hire you without one - such as social work. I have a friend who works in this field with a bachelor's and has done very well, but her organization pays new employees more if they come in with a master's degree. When thinking about this pay difference, be sure to also factor in the cost of the degree itself! Unfortunately there is no perfect formula to figure out what makes the most sense financially - some experience in the sector will help you get a better idea of salaries to make an informed decision. (You can also learn about nonprofit salaries using resources like this survey data from Third Sector New England: http://tsne.org/valuing-our-nonprofit-workforce-2014/compensation-data.)
Most of the jobs that I have seen which require a masters degree are high-level program manager jobs or department director type positions. For example, at my last job we had program directors for program areas such as the human right to water and economic justice, and those people really needed a masters degree (or law degree). Beyond entry level positions, a master's degree is always helpful, but only sometimes required.
Finally, I recommend you spend some time looking at jobs on sites like idealist.org to get an idea of what jobs are out there and what the requirements are. If you are interested in a particular area, such as program management or fundraising, definitely search for jobs in that area and read some job descriptions. This will help you get a sense of what education would be most helpful for your future plans, as well as what tasks are involved.