I got my private in a Katana.
Fun airplane to fly if somewhat limited in its usefulness.
My personal observations (I've also flown 172s, a Warrior and an Archer)
Pros:
* Great visibility
* Light controls
* Very lightweight
* Constant speed prop
* Modern avionics as standard
* Throttle quadrant
* Rudder authority
* Slips not prohibited
* Spins not prohibited
Cons:
* Very lightweight (you get bumped around a lot more than in a 172)
* Light controls (only really need to trim in cruise)
* Automatic mixture (something a pilot transitioning *from* a Katana needs to learn)
* Cramped cockpit
* Little useful load
* Stick is fun but I spent my first few hours in a 172 trying to steer with the wheel on the ground (t*sser I may be, but I'd never had the temptation to steer with the stick in the Katana)
* Slow (DA20 is anyway)
* Horizontal metal bar near right toe brake (LHS) means that you may be furiously pressing the bar instead of the right brake. This is dependent on how you've adjusted the pedals.
* DA20 is NOT IFR CERTIFIED. However, most have the instruments you'd for IFR practice, but you wouldn't be able to go into IMC.
Out of what I've flown, my favourite is the Archer followed by the Katana. For whatever reason, I'm not keen on Cessnas. Low-wing vs. high-wing and the fact that I learnt in a low-wing? Maybe - who knows.
There's a Katana with a Continental engine (C1 I believe), and that gets over the speed and mixture points listed above. The new DA40 is a four-seater and now has both JAA and FAA IFR certification, and looks very nice. Comfy too - I sat in one at an exhibition and have heard first hand flight reports (both front and rear seat occupants).
You can definitely stall them, but they certainly don't break sharply - somewhere between a Cessna and a Warrior I would imagine. They do, however, encourage sticking to the POH numbers. Any excess airspeed and you will float forever, and they like to fly - took me ages to figure out how to land rather than arrive simply because flaring too early is going to take you quite a way back up relatively speaking (especially if you've got some excess airspeed).
Soft field take-offs are great fun. The nose really pops up and if you're not expecting it you will bash the tail on the runway (so not fun if you're not prepared...)
All in all, I think they're great fun to fly. Feels like you're in a sports car, but a "toy one" like a Mazda MX-5/Miata rather than a Porsche.
David.