Hey let's go chuck a flatball!
I'll generally drop whatever I'm doing and go play frisbee if at all possible, and I usually end up giving tips on form. Most people make the same very small set of mistakes, and with a little attention, progress is very fast. There are just a very few things to focus on, and while thinking about them puts you in the wrong part of your brain to do them right, once you realize the concepts, your body will start to pick up on why some shots work and some shots don't, and you'll learn fast. Frisbees generally respond pretty directly to what you do to them; by examining what went wrong with the trajectory, you can tell what went wrong with the shot: if it arches up the left and then dies to the right, your arm went too high in the middle and then too low at the end.
Let's start with the basic backhand. First, line up your feet. Your torso and toes do not point at your target, but 90˚ to the side. A line going through your feet would point to your target. Bend your knees and put your weight on the back foot. Cross your throwing arm to the back, and cross your other arm above. If your other arm just hangs limp, your spine tends to rotate wherever; making it an active component really helps stabilize things, so trust me and do that. Think of a tabletop, and keep the frisbee on that: starts flat, stays flat, same height, releases flat, same height. Any diversion from that will be reflected perfectly in the path of the disc. As you toss, shift your weight to the front foot, but do not extend the front leg, or go past it, or lock the knee: keep it soft. Straightening that front leg is the most common problem; fixing that makes a huge difference, right away. That top arm, whip it around to the back, palm open, as the front arm unwinds; that too makes a really big difference. Basically, it's just a lateral shift and opening: cross your arms, weight on the back foot, and as you shift your weight to the front foot, open them together, spine vertical, shoulders relatively square. It's a kind of a whip, first the hips, then the shoulders, then the arms, the wrist. You'll feel that when you get it. It's also not necessary to look directly at your target; just locate them in the corner of your eye and let your vision go soft.
I generally try to exhale at the release, and purse my lips so I have some backpressure to focus against, really helps you concentrate your energy, but without popping a vein. Don't try so hard, it's just a damn frisbee. Who cares.
One thing that really helps stabilize shots is learning how to put more spin on it. That's not something that lends itself to verbal explanation very well, but it has a lot to do with making that wave of snapping go all the way down to your fingertip (which goes on the side of the rim, not under), which sometimes, right at the end, actually goes back in the other direction. For starters, think about cocking your wrist and only releasing that last angle at the very end of the toss.
The hardest catch is two hands, one on either side; one hand usually knocks it out of the other. Just don't do it. The easiest is pancaking it, one hand on top, one on the bottom, just clap them together flat. If the frisbee is arriving at chest level or below, catch it palm up, with the fingers on the bottom. Try and reach out towards it, and pull it in towards you, rather than just snag it at a point. If you're right-handed and it's coming low on the left, try and get the back edge of it; you can some really fun swoops that way. Those are also fun if it's just above your head (thumb under on those).
My favorite style is to try and turn the catch into a toss in one smooth move, just flowing the momentum back into a toss, before taking a third step. For example, if you catch a trailing edge on the left, continue the rotation around and pop a forehand toss on the right, or uncork that wound-up motion back into a backhand.
The other main shot is the forehand. Make a V with your fingers; the side of your middle finger closest to the pinky goes against the inside edge of the rim (easier to have someone show you). Face the opposite way from a backhand position. If you are right-handed, this shot will go out a little ways then bank 90˚ and dive on its left, every time. So, you strongly overcompensate until you work out the weird wrist flip that makes it work. Lean way over to your right, bend your head way over, leave it there past the follow through. Your other arm doesn't cross; point it towards your target. Cock the frisbee up behind you, angle the path down towards the ground, and release high, like a big slice. Basically just take that whole flat level plane you were trying to do on the other side, and tip it right over. Once you can get the shot to arrive flat, you can start trying to work out that spin, but if you start coming up and it starts going down, then lean right over again. Once you get it you can really wing this one, and with just a compact flip, not a big arm move.
If you can find some nice thick grass without a lot of rocks and sticks and stuff, you might want to explore diving. Start out close to each other, and toss them a foot or two off the ground way off to the side, and try to land fully extended. The key is keeping your elbows and knees out of the equation; let your energy go straight out towards the frisbee, and land on your torso and thighs. You'll want to reach out and break your fall, but doing that directs all the force into your joints (small, away from your center of gravity, which leads to twisiting and wrenching), instead of onto a broad flat surface, which might get scraped up, but that's better than broken or sprained, and can actually put a big smile on your face: blood! Relax, let the air get forced out when you hit, and don't chop your tongue off.
There's a most excellent field sport called Ultimate Frisbee. If you look around, you can usually find pickup games in your area, and they're often co-ed. There's also occasional tournaments here and there, which are great fun, especially the women's teams; they really go for it, and often work together as a team better than men, and the hips thing often makes for fierce lateral quickness. The sport itself is kind of unique in that sportsmanship is a major priority; you just don't foul, and if you do, you can call it on yourself and nobody on your team will think you a traitor. Games don't even usually have refs: think about that. Game play is pretty straightforward: catch the disc, you stop running, catch it in the endzone, you get a goal, one point, and beginners are the ones that catch all the goals. Basically, keepaway, but with endzones. Huge fun, and a great place to meet mellow smart people.
Gimme call any time, love to toss.