You're not gonna become a master in your spare time, but you can get decently good at photography in five years of dedicated but non-full-time practice. It's taken me about twice that to get decent, but I fucked around a lot along the way.
You don't need a ton of fancy equipment, but photography does require some gear. I'll limit the gear-buying here to once a year (not counting books).
Start by buying a digital SLR. Mirrorless systems are neat but it's just not the same as looking through a lens. You don't need a top-of-the line camera here -- the entry-level models from Canon/Nikon are fine (if you can afford one step up from the entry-level you'll save yourself a bit of frustration, but it's not critical). It doesn't matter which brand you pick. Used gear is fine if it's from a reputable seller.
Get a zoom lens to go with it (a 18-70mm would be ideal, but a bit in either direction is fine). If you're buying new this will be bundled with the camera and called the "kit lens" and it's fine for now.
Get a fast 50mm prime lens too (at least f/2, preferable f/1.8 or f/1.4 if your system has one). You want this because it's going to be a lot easier to understand depth of field when you can see it more obviously.
If none of those words/numbers make sense just ask me and I can point you at what to get.
Total cost will probably be around $900 if you buy new, or a LOT less if you buy used. You can delay the 50mm purchase for a while if necessary.
On to the actual learning. For the first year you have three assignments.
The first is to take at least one photo every day and post it online (flickr/whatever). Look up "365 photo challenge" for lots of examples of what other folks have done. You need to post it online to keep yourself honest. This will force you to practice and motivate you to get better.
The second assignment (done concurrently with the first) is to learn how to use your camera in full-manual mode. This will be a pain in the ass, but it's worth it (just be glad you don't need to learn to use a darkroom at the same time).
There are a billion photo books out there that teach the basics. One that's free online is the Bastard's Book: http://photography.bastardsbook.com/
About six months in you should have a decent understanding of exposure and a few other basic topics. It's time to learn how to process photos. Pick a photo processing application like Photoshop or Lightroom. My recommendation would be Lightroom (now that Apple has killed Aperture I think it's the only sane choice). Buy the software and a book about how to use it and process all the photos you've taken so far.
Take a LOT of pictures this year. Aim for at least 5000 total (not all keepers of course).
Now that you know how to work with a camera, you need to learn to work with light. Spend this year reading everything on the Strobist site and doing all the exercises. Your gear buy will be whatever lighting equipment Strobist recommends at the time. So a flash, stands, umbrellas, pocket wizards, etc.
Now that you're done with the 365 challenge, do another 365 challenge, but make them all self-portraits. This will help you get comfortable in front of the camera, and ensures that you always have a human subject for your Strobist homework.
If you run out of Strobist homework, buy Light: Science and Magic and work through that book.
Take at least as many photos this year as last year (5000), preferably more.
At this point you've taken over 10000 photos, which is a good start. Now it's time to slow down and be a bit more careful. Try to photograph once or twice a week, but plan each outing carefully. Planning makes a big difference.
Your gear this year is a tripod (disregard Ken Rockwell's bad awful advice -- if you're serious you should have a tripod). You can spend $300 or so on a big heavy aluminum one, or $1200 on a Gitzo/RRS carbon fiber combo. Those are your choices: anything else (e.g. super cheap aluminum) is trash.
The aluminum one will work and will be stable because it weighs a fucking ton, but it'll be a pain in the ass to use. The carbon fiber one will kill your wallet but unless you run over it with a truck it'll last your entire life.
Decide what "genres" you really love. Landscapes? Portraits? Street photography? Buy a couple of books about the specific topics you're interested in and do all the homework they give you.
Also start scouting around your own town for photo meetups and such to meet other photographers. Start making friends.
If you've made it this far, go ahead and give into temptation and buy a fancy new camera. You've earned it. Also get a nice camera bag (Think Tank Retros are great) and a good strap (I like Black Rapid).
By now you'll have outgrown the "Digital (Landscape/Portrait/Street) Photography" kind of books because they all say pretty much the same thing. It's time to move on learn from the masters.
Step one is to find some photographers whose work you really like in areas you're interested in. Ask your photographer friends for advice. E.g. if you shoot portraits, find Gregory Heisler. Landscapes: Ansel Adams, Galen Rowell. And so on.
Try to find any books by those people that are basically a collection of their photos and them talking about each one (preferably at least a page or two per photo -- a single line caption doesn't teach you anything). This will give you a glimpse into their mind. Examples from above:
- 50 Portraits by Gregory Heisler
- Examples by Ansel Adams
- Mountain Light by Galen Rowell
You can also try to find workshops and/or videos from folks whose body of work you admire. I've begun working through Ming Thein's "Outstanding Images" videos and am really liking them so far.
The key thing here is to find people whose photographs you love. There are plenty of people out there whose names are at the top of Google and Amazon searches because they're great at marketing themselves (Kelby, Rockwell, etc) but you want to find the people whose photographs really speak to you, not the ones that are best at SEO'ing Amazon and Google. The work comes first, always.
I think that's it for now. Let me know if you have any questions. Good luck!