So I had a big long reply all typed up and my laptop just deleted it. Poo.

You can buy the cups at Walgreens, CVS or Walmart.
For me the cups are a miracle. As someone who's used pads and tampons too, I definitely prefer the cups. While they go in the same way as tampons (and do require some pushing with your finger) I can't feel it when I go about my day. I was always painfully aware of the cotton plug, but after the first 5 minutes of putting in a new cup I forget it's there. It locks in under your cervix in your pelvic bone - sounds scary but it's not that bad, all you do is push it in until you hit the back of your pelvic bone and once you stand up it'll adjust itself. There's also no menstrual smell with it, I always experienced that nasty lady-musk when using tampons, it's totally gone with the cups.
The only time the cup leaks is when I'm using the bathroom - and the leaking is only due to bearing down, once I stand up it readjusts itself and no more leak! I've experienced tampons leaking and being rejected either because I didn't insert them far enough or because I was moving around too much that they just started to work their way out. Also, if you leave the tampon in for too long you may leak onto the string, which can get kind of gross.

Also, remember that tampons can cause Toxic Shock Syndrome if left in more than 8 hours - while it's rare it is still a possibility. It's not an issue with the cups.
Disposal is similar to any other menstrual device - take it out, wrap it in TP, toss it, and put in a fresh one. The brand I use is 7.99 per box (which has a 7 day supply) - while it's more expensive than tampons (which are like 6.99 for 40) it's definitely worth it for me.
@jmaxley: From my experience with the cups I can't see how that could happen, but that may vary by brand. I use the Instead brand, which is a soft, flexible cup. There are a few other brands out there that are harder cups, so I could see the possibility of an issue arising with those.