Each person's career path is different! There are artists working in the animation industry who have completed multiple Bachelor's Degrees, and others who have either dropped out of school or have never pursued education past High School.
It might be helpful to get to know yourself first and start to understand how you learn best (while also considering time and budget for education). And of course, try not to underestimate the amount of resources available to you online! There is plenty of free information out there to help you get started.
I'd also recommend taking advantage of internships while you're still in school. Sometimes you can work your way up through a company by starting off as a Production Intern. At the very least, it can be a wonderful learning opportunity to better understand how Animated TV or Feature Productions are run!
You can start by exploring the wide array of positions available in the animation industry (or through your own Independent work), discover what part of animation you're interested in, and find where that leads you!
Megan recommends the following next steps:
It is hard to say really. It takes a lot of practice and varies from what you wish to accomplish. Do you want to be a 2D animator or 3D animator? There are a few steps you could take to get you started.
Michael recommends the following next steps:
A four year bachelor's degree would be a great way to get a job in the industry. It never hurts to start practicing early though! You can start animating before college and play with some apps on your phone. If you are interested in making cartoons, read "Understanding Comics" by Scott McCloud. Study your favorite cartoons. Animation and cartooning are sequential art forms, meaning they rely on time- one thing happening after another. Learning how to tell a story with your art will help you become a better animator and cartoonist.
You can also get your master's degree in animation from schools like California Institute of the Arts or the Royal College of Art. This will take six years of education.
There is also the other type of cartoonist like Stephan Pastis - I recommend reading his wikipedia article- who followed another line of work, which he was very unhappy with, before becoming a professional cartoonist. He spent 9 years doing his other job before he was able to break in to the professional circle, but it is a great example of an alternative way to becoming an established artist.