Well, for 'real' cartography, I'm assuming you would approach it like any other discipline - that is, taking courses either in person or online, 'apprenticing' to someone who's already in the field, etc.
For fantasy cartography... it's a bit more problematic. I suspect that those of us who have been doing it for a long while started out scribbling on scratch paper as kids and have just upped our game bit by bit since then. I know that's the way I started out, and from what you've said, so did you. I don't think there's any hard and fast starting point or travel map. Do what thou wilt. I would say take a look at where your interests lie. Are you more intrigued by the idea of mapping a whole world and then mapping out progressively smaller and smaller areas? Have you wanted to try a city map? A cave system? Start with what interests you. Start with something that's finishable. Don't bite off more than you can chew and get discouraged and never map again!
If you want to work on an entire world, what is it in service of? Are you writing a novel? Is it for an RPG? I would assume you already have some ideas about what you want your world to contain - vast deserts, rhino-riding orcs, who knows? Creating fantasy worlds (and maps) is one of the most rewarding creative experiences you can have in my opinion, but there's so much to it that you could literally write a book about it. And people have. What I'm saying is, the purpose of the map/world should probably dictate how you go about creating it.
Really, like most other art, the best way to do it is to just do it. Draw some sketches. Play around with the tutorials here. Experiment, don't be afraid to post works-in-progress and ask advice.
Hopefully that ramble was of some use.