First, about the handwriting. I have mediocre handwriting--it's a legible (to me, most of the time) scrawl. This is because I don't care about my handwriting all that much. I'm sure there's some minimum neurological competence you need to do calligraphy, but I bet anyone whose hands are in good enough shape for ordinary living has got it.
This "I can't/won't try calligraphy because I have bad handwriting" thing leaves me comprehensively pissed off at conventional schooling--that's where people get the impression that if you didn't learn school subjects and topics related to them in school, you can't learn them at all. This is a false idea, and utterly poisonous.
OK, back to calligraphy, a much more pleasing topic. It's certainly possible to be self-taught. I was--I worked from books (probably the Speedball book and Johnston's _Writing, Illuminating, and Lettering) and
copied out the second two books of the Divine Comedy into notebooks, using various lettering styles. A casual look at calligraphy books suggests that they're generally pretty good and cover similar points. A major reason to accumulate calligraphy books is admiring the calligraphy and wanting the different styles.
A teacher might have helped, but what I had was something I liked enough to be willing to put in a lot of hours into. I might have been able to learn faster if I had more conscious goals, but the truth is that I just tried (and still try) to make the letters more satisfying when I letter. A lot of this is a hunt for eveness, but it somehow brings good proportion and smooth strokes along with it. I've been lucky that my ambition is always just a little ahead of what I can actually do. I'll look at my ten or twenty year old work, and be amazed that I didn't notice how bad most of it was, but it's probably just as well that I couldn't see it.
Some books recommend starting out with projects as soon as possible so as to get into the habit of doing design. This is not a bad idea on the face of it, but I don't know if I would have done as much calligraphy if I thought I had to do a lot of design, too.
Materials are important. I recommend going into the hobby with a little willingness to waste money, since sometimes when you just can't get a decent stroke to happen, it's because there's something wrong with your combination of penpoint, ink, paper, and/or temperature and humidity.
I started out with Osmiroid and Platignum fountain pens, and went on to dip pens. The big advantage of dip pens is that you can use absolutely opaque inks, and I love absolutely opaque ink. I currently use Brause points and Tape points--these are recommended for the heavy-handed calligrapher. If you want a springier point, try Mitchell. Tiny little detail not known by the general public: Brause and Tape 1 mm points are different sizes. This is good news, since it gives a little more flexibility in stroke thickness. Don't tell the companies, or this "problem" might get corrected.
Speedball (the most commonly available points/nibs) aren't bad. I've seen very good work done with them. Sometimes I wonder whether I've defaulted to more obscure points out of sheer snobbery.
And if you want to get started the easiest way, try felt tip points. Zig makes very nice ones in hundreds of colors.
These days, I'm using Moon Palace Ink, but I'm not completely in love with it. I may need to waste some money in search of a more opaque and cooperative ink.
I use gouache (opaque water colors) for color. I've used Windsor and Newton and Schminke (both companies also make good metallics), and McCaffery's Ivory for white.
My two suppliers are John Neal Books and Paper & Ink Arts. They're both very good, and have product lists that don't entirely overlap.
I use plate bristol board (the smoothest) for button masters. Cottonelle or something of the sort for white, Coloraid for solid colors and black (this is paper with the colors silkscreened on--it's a wonderful texture/surface, but a little expensive and has to be mail-ordered), banner paper (also for solids--cheap, good surface, but not very many colors available), and "vellum", which is very nice to letter on, but unforgiving if you make badges out of it, not to mention various multicolored papers designed for laser printers.
Calligraphy is kinesthetic/tactile as well as visual. It's important to feel the contact the pen point (nib, except that I have a habit of calling them pen points) makes with the paper. In particular, the basic strokes require that you keep the whole edge of a broad point in contact with the paper. (Sometimes you're using that whole width, sometimes you're getting a hairline by moving in the direction of the width of the point, sometimes you're getting varying width of stroke because you're intermediate between the two directions.) And if you're at all interested in T'ai Chi or anything of the sort, I recommend it. What you can do with your hands is influenced by your general coordination.
On the other hand, I've had people tell me I look like I'm meditating when I letter. Part of me might be meditating, but the conscious part is probably worrying about money, trying to figure out how to deal with hypothetical slights (really! and I have no idea why I spend time on that), or contemplating dinner.
I found that I could letter more evenly if I expanded my vision/attention to include what I'd already written. This makes it easier to have a consistant size and slant than if I'm just depending on my mental image of the letters.
And if you've got questions about any of this, just ask.