T_Richard,
If you want to take the controls of an aircraft, you can do that at just about any flying school. What you're looking for is a "trial lesson" or "trial flight". It'll probably include either 30 or 60 minutes of flying something like a PA28, C152 or C172, together with about 30 minutes on the ground with the instructor before you go. Just pop down to your local airfield and ask around, or look in the Yellow Pages. There's no obligation to take things any further. One word of caution, though - many people have done this, not intending to take it any further than the trial lesson, have become instantly hooked, and have found the time and money which they didn't think they had to go on and get the license!
On the other hand, if you're looking to fly an old or interesting aircraft like the Harvard, you'll need to look a bit further. These flights are normally billed as lessons, they have the same legal status as a lesson, and the instructor will have the same qualifications as the instructor at your local flight school. But you might find it's slightly less hands-on than a genuine trial lesson - that's because it's aimed more as a birthday present for grandad than as a sample to get you hooked. This is, of course, a generalisation, but if you want hands-on flying and an interesting aircraft, then make sure your instructor knows that before you take off.
Have fun!
FFF
---------------