Your hunch is correct. A fully geared folding bike like a Dahon (Boardwalk or Speedster recommended), Brompton, or Birdy seems appropriate for you.

This way you have the option of building up your fitness and always have a way out (take a mass transit w/ your bike folded) if you get a flat and you don't know how to fix a flat yet, when you get tired, or if it rains hard. Also because the bike is geared and comfortable, you can use it to increase your mileage gradually and take it on the train/bus, when you fatigue sets in. It (Dahon) also has lots of accessories for hauling gear and change of clothes.

If you're commited to using a bike for short distance shuttles (less than 5 miles each way), like from your house to train station and from train station to office, I recommend the Strida, which is almost perfect for this purpose (better than other folders because it's easy to carry and folds friggin' fast: I fold mine in less than 5 seconds while moving). The only problem with Strida is that it's not suitable for longer distances (no gears, but has freewheel).

I started out w/a Strida and eventually bought a full-sized commuter. If I had to do it over again, I'd probably start out with a Dahon Speedster. I keep the Strida at work and use it for running city errands because it's virtually theft proof (just fold it and take it with you), not that bike theft is a major concern in Japan, but it's a major concern here in NYC.

Good luck and glad you looked into cycling as a solution.