Knobby,
I think you're probably at just the right time to start looking into it. My PPrune name means what it says. I only have 120 hrs of Group A time logged in the two and a quarter years since I started my PPL course (on a Piper Warrior), and had 60 hrswhen I started on the Cub, about 85 when I started on the Yak. Must admit to about 40 hours of gliding in a previous life plus some highly informal unlogged time with some very good ex RAF / airline instructors in various types during the 80s, but it's not a tremendous amount.
Don't be afraid that people will reject you for lack of hours. Everywhere I've talked to about flying different types, the attitude has been "We'd love you to have a go, don't worry, we won't send you off solo until you're good and ready". What counts is attitude and aptitude more than hours. Where they might express a view is in encouraging you to take things a step at a time, e.g. C152 to Cub, Cub to Tiger, Tiger to Pitts, not straight there in one fell swoop. But I've heard of that being done too, by the truly determined, though I reckon it often costs more and takes longer than going via intermediate steps. In any case, many schools with more exotic types are just as happy for you to have an hour or two's dual taster of a type, even if you don't want to solo it.
What really, really helps you be accepted as serious about learning to fly a type, in my experience, is reading absolutely everything you can about it. Get the pilots notes or flight manual and learn it off by heart. Understand the engineering and the systems. e.g. learn how a CS prop works, and why (for example) a lot of aerobatic types and twins have a type of CS prop that works in a different way to an SEP tourer. Read lots of books by other pilots who can communicate well, especially (if you can get them) the 70s and 80s writings of Neil Williams, Brian Lecomber, Bernard Chabbert etc. These days I find Andy Sephton and Maxi Gainza to be among the best writers about the foibles of various types. And ask your instructors, and people whose flying you respect if they've ever flown a Type X, adn what they thought about it. Often the quiet softly-spoken guy who does a bit of weekend instructing turns out to be a former Lightning or Harrier man who flies corporate turboprops during the week and displays a Stearman (or something) when he can.