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sha90 "Mamachari" is the Japanese version of the 'Dutch Bike' or city bike. Mainly different in that they tend to be small-wheel bikes, since in Asian urban environments, space is at even more of a premium than Europe. Either way, they tend to be simple, rugged bikes, usually 3-speed, upright, and overbuilt, since they spend most of their lives on outdoor bike racks. They are relatively inexpensive, because they're a tool, utilitarian, kind of how most Americans view something like a lawnmower. You take it out, use it, and put it away, and don't think about it until you use it again.
In the US, this type of bike is not easy to find. There are some, though; Specialized's Roll, and the Electra Townie are relatively easy to find, and while they've got derailleurs, it's as common, and low-maintenance running gear as you'll find on any decent bike. They're good bikes, and though inexpensive, they're not cheap. ($500-$600)
Another option, if you really really want a small-wheel bike would be to look into a folding bike like a Brompton or Dahon. Most of the features you'd find on a 'big' bike, but built around (usually) 20" wheels. Folders are good for short-distance hops, and because of the way they're designed, are good at fitting small adults, or big kids who aren't quite big enough to handle a full-sized bike.
If it's 'just' commute and run to the grocery store, that doesn't sound too demanding, and a typically decent (bike shop level) bike would be able to stand up to it for years without needing much. Would also help if you could share how far of a commute and what metro area, since some places require a little more out of the bike than others (say Seattle / PNW)