Start with studying for the CompTIA A+ exam. While it's not particularly relevant to networking, it's a good idea to at least have a fundamental understanding of how computers work before you get into a more specific technology field like networking.
If you think you're already there, or just want to jump straight to networking, start studying for the ICND1 (CCENT), and then the ICND2 (CCNA). You can use books, instructor-led classes, or online video training. If you go the video training route, I'd still suggest picking up a book or two.
Popular online classes include Wendell Odom, Todd Lammle, Chris Bryant, and Lazaro Diaz. You can find some of them cheaply through a Safari Books Online subscription ($100/year with an ACM membership, and you'll also get access to some books) or various online training places like Udemy.com (typically around $10 per class) or Academy.GNS3.com (about the same price).
To expand on my comment, if you want to get into the networking field, you basically want to fully understand all topics covered in the ICND1 (CCENT) and ICND2 (CCNA) exam blueprints.
These are Cisco exams, but the topics covered are largely industry-wide. Things like routing, switching, subnetting, etc. Even if you never touch a Cisco product again (unlikely), you'll be served well by understanding these topics.
Once you get the basics (CCNA Route+Switch), you can figure out if you want to go deeper or wider into the Cisco ecosystem: Deeper (CCNP R+S and CCIE R+S) or Wider (CCNAs Security or Wireless or Collaboration or Service Provider, etc, most with their own higher-level CCNP and CCIE certs). Otherwise, once you get the fundamentals, you may want to branch out into other vendor certs if you are working with those technologies (Juniper, etc).
How long did it take you to fully grasp it?
Not OP, but I feel that people make subnetting out to be more difficult than it actually is. Once you learn how (which can be done in less than an hour), it takes a little bit of practice to get better at, maybe like a couple of practice problems a day for a week, but really anyone can do it.
This. Subnetting is "Easy" but getting good at it to be very fast in your head is the "hard" part.
Once you're at that point it's just practice, practice practice!
I was going to post this officially next week... but I went ahead and released all five videos of my subnetting mastery series. It is really all you need to become well versed at subnetting.
Udemy.com, for 10 bucks you 1000s dollars worth of networking knowledge. I was A+certified before getting my ccna, A+ is worthless and expensive. Just go stright to ccnax or ICDN1
networking for dummies then A+, Net+, CCENT. networking for dummies is almost enough to get you through the Microsoft cert - networking fundamentals.
Figure out the difference between a switch and a router, then LEARN THE OSI MODEL. It might seem boring but it's absolutely invaluable. You absolutely must know the OSI model.