How do I reliably connect power, ground and signal to a remote device?
You want to look at connectors with "ip67" protection to fully protect against dust and water immersion. If all you need is "resistance" to dust and splashes, "ip54" is about as low as you'll want to go. (Wikipedia on IP ratings) Note that the ratings assume you assemble cabling correctly with the right diameter cable, heat shrinking, possible gaskets, cable sleeve rating, etc.

These connectors will then come with some number of pins -- choose as many as you need (from 3 and up.) Also, the pins will have current ratings; choose a current rating that is suitable for your power needs. Finally, you need a mating set of male and female in both ends.

Many applications use round connectors (think "XLR" from audio) with screw rings to securely tighten the plug to the socket on each end. You can start looking on Digi-Key for ready-made cables, if you don't want to be poking connecting pins into sockets yourself before you start soldering...