Internships are really the most important thing you can do to gain experience in the field and eventually land the kinds of opportunities to which you aspire. Nothing beats hands-on experience and an understanding of the daily goings-on of a gallery or museum setting, and internships--generally unpaid, unfortunately--are the way to go about that.
Some museums and galleries have structured internship programs, but you'll find that most don't. I secured most of my internships by emailing persons of interest at local galleries and museums and inquiring about opportunities. Most of these institutions and organizations could do with some volunteer help, so I had a lot of luck with this approach. It helps to be proactive.
Education is also a big part of securing certain kinds of positions. If you're interested in curatorial roles or certain kinds of administrative positions, then you'll likely need to pursue a graduate degree in either art history, curatorial studies, or museum studies. The kind of degree and specialization very much depends on the kind of job you eventually want to do.
I consider myself something of a success story. I was hired as an assistant curator in contemporary art in the last year. I have my BA and MA in art history, a very niche but upcoming specialization, and I did a good number of internships before I transitioned to contract work and then full-time roles. I didn't have any fall-back plans. If you're hardworking and tenacious, you will find more than enough opportunities.