We bought a Sony Alpha series camera ahead of our first daughter's birth in 2009 to replace a well-worn Olympus OM-10 that I'd nabbed from my dad.
We loved it, picked up various extra lenses and even an extra body but found that as our knowledge and technique improved, the camera itself showed up its limitations. We've now (in the last month) gone for a Canon 6D - which I think is the cheapest full-frame body that Canon does - we managed to pick it up for about UK£750 secondhand, body only.
Everything (lenses, flash, accessories) is more expensive but the step up in quality is astounding. If I could go back in time, I'd buy a full-size sensor camera right from day one. But if you're just dipping your toes in the water, those Sonys are great value, flexible - and easy to use too. Although one issue we hit was that very, very few places hire out lenses for them so you might struggle to check out new glass. If you're willing to go secondhand, I bet you could get an A200 and a couple of lenses and be well within budget.
btw I'm not sure if the taping analogy holds here. When I had only film cameras, a 36-exposure roll could last me a month or more. Nowadays, I'll easily hit 200 snaps a day - at least half of which are duff/testing settings/indistinguishable from the shots either side but the number of great shots we get has gone up dramatically since the days of film.