Flying Lawyer and AfricanEagle have both made eloquent points, and I agree with everything they've said. Flying a taildragger WILL make you a better pilot both on the ground and in the air.
On the ground for obvious reasons but some not so obvious, as well. Because most of them tend to be high wing and lightweight, they are very susceptible to winds while taxying. They will teach you to sharpen up your wind awareness on the ground, which translates well to any other aircraft. Because of this, they have to be flown all the way to the hangar, not just to the touchdown. You can often identify a poorly-trained nose-wheel only pilot on landing because as soon as the nosewheel is on the ground, you'll see the elevators droop as the pilot instantly drops the backpressure because "the flying is done". That's not great technique in most airplanes, but it is essential to get it right in a taildragger.
They may teach you to be a better pilot in the air, too because most of what you will have available to you is likely to be vintage stuff - ie no systems, bugger-all in the avionics department, low-powered, with big, long wings and loads and loads of adverse yaw. Training in a C152 or a PA28 will not give you experience with those. You'll learn how to use the rudder to counteract those fistfuls of adverse yaw (not true in all types, but more likely than spamcans as a class). These traits will all teach you to manage your energy better, handle your controls much better, and to learn to fly with all of your senses and your head out the window ('cause there's nothing worth looking at inside!). After a few hours you'll be able to know your engine RPMs by sound, your airspeed range by the feel of the controls, your height by the size of the trees (if they're too small, you aren't low enough!), and your heading by the position of the sun. OK, I'm kidding on that one, but you'll find flying a much more visceral experience that pulls in your whole body to paint a complete picture.
As for specific types, the Stearman is the class leader. It's big, it's loud, it's brash, it's...well, it's American in every bombastic sense of the 1940 definition of the word. And it's lovely to fly. It handles like a big Cub and open cockpit flying is something that everyone should experience as much as possible in life. That is one airplane where the nostalgia is entirely appropriate. I remember when a newly rebuilt one showed up at Leicester one afternoon, all shiny and clean, and a bunch of us went out to look at it parked on the ramp.
"Will you look at all that stuff hanging out in the breeze there?", one chap commented. "All those wires and the fat landing gear and struts and stuff. There must be a hell of a lot of drag".
pause while we all considered that comment.
"Yeah", someone finally said. "But it's VINTAGE drag!" General agreement all round that it was, indeed, glorious, golden, vintage drag.
Can't comment on the Stampe except by reputation, and all of which I've heard matches with FL's post.
I completely concur with FL's assessment of the Tiger Moth (a great airplane to have flown, much like Scunthorpe is a great place to be from). However, I really don't think enough has been made in these posts of the true horribleness of the Tiger Moth. Seriously. It's nastiness as an aeronautical abortion has only just been hinted at in these pages. I can't say "words cannot describe" because there are words that can describe the horrendous hideousness of the beast, but none of those words are appropriate for a family show. Really. There should be a special place in Hell reserved for the designers (we can't call them "aeronautical engineers" because they clearly showed that they weren't) where they have to spend eternity trying to make coordinated turns and trim the bastard, all at the same time.
As for the Harvard, it's clearly the next evolution of the Stearman and they make a perfect pair. Once you've got a fair few hours of tailwheel time, go try out the Harvard. WITH tailwheel experience, you can concentrate on the issues of size, power, complex systems, and mass. Your piloting skills will also be much sharper, so your ability to cope with the new things will be much improved. WITHOUT tailwheel time, you'll lose track of which is which as you try to get the hang of the little round thing at the back as well as all the other issues. My opinion, anyway.
But it's not a fire-breathing warbird that will take you tearing across the sky in blazing glory (not until you taxy up in front of the crowd at the restaurant, anyway
). It is, after all, a trainer, and is sort of the C152 of its day - it's stable, reasonably benign, and not terribly fast or powerful. Its purpose was to teach instrument flying, formation, and complex systems and, for its time, it was magnificent at all those. In many ways, the experience of flying it today is about the systems and job of the airplane more than its performance. If you want true wardbird flying, save up your pennies and head to Stallion 51 in Kissimmee and get an hour in a Mustang.
Anyway, pontification closed for the afternoon. Time to go grab a beer.
Cheers!