Is there a standard tripod mount? Is it reasonable to assume that a given tripod will attach successfully to a given camera, or will I have to research my options?
Yes. The tripod thread is standard: 1/4-20, which means ¼", with 20 threads per inch.
This is specified by ISO 1222:2010. I'm not willing to pay the $57 for my own copy, but I'm kind of curious, as wikipedia says that the current standard also allows 3/8-16 — apparently that's an older mostly-European standard. This is probably old-hat to aficionados of classic field cameras, but was new to me — all of the modern Japanese DSLRs and compacts I've seen use the 1/4-20 thread. 3/8-16 may still be common for larger-format cameras. I checked my grandfather's Voightländer Bessa medium format camera from the 1930s, and it uses 1/4-20. I think it's safe to say that for consumer and mass-market professional cameras, 1/4-20 is universal.
The 3/8-16 standard is in wide use today in photography, though — just not for camera mounts. It's common for lighting gear, including lighting stands and mounts. I have some Manfrotto gear with the reversible studs — basically, camera mount one side, lighting equipment the other way. Also, as Michael Clark notes below, 3/8-16 is typical for connecting tripod heads to the legs.
The other important thing is that almost all modern tripods use a quick-release plate system. There's a small plate which has the tripod thread which screws directly to the camera, and then that snaps into the tripod head using a proprietary-to-each-company attachment. That means that even if your camera would use the less-common thread, you could get a plate that matches — for example, a Gitzo plate which comes with both threads. And adapters between these two threadings are readily available — probably mostly for ease in mixing and matching between camera and lighting support.