Joe - Standard tuning nowadays seems to be re-entrant C6 or gCEA, though some folks still use the re-entrant D6 or aDF#B. Some, like Helen suggested, use the linear C6 or GCEA, like the first four strings of a guitar capoed at V. Baritone ukes are tuned DGBE. Nylon strings are nearly always used.

I prefer the re-entrant tuning, with the 4th string raised an octave. There are songs that take advantage of this raised 4th string. I play a version of Pete Seeger's "Living In The Country" that can only be played using re-entrant tuning. I am also a clawhammer banjo player, and the 5 string banjo is also a re-entrant tuned instrument. Clawhammer style or up-picked style ukulele realy works well on ukes with a high G.

As Helen pointed out in her post above, the re-entrant tuning does rob you of G,G#,A and B, restricting your melody playing somewhat, but it makes for some really nice close chord voicings.

Are there other instruments that use the same tuning? A tenor guitar in "Chicago tuning" is tuned like a baritone uke. Nick Reynolds of the original Kingston Trio played a tenor guitar, but usually had it capoed at the fifth fret, making it a linear tuned ukulele with steel strings. He was originally a uke player, so this reduced the learning curve.
The Tiple is also tuned similar to a uke. My tiple is a tenor uke sized 14 string instrument made by the Regal company in Chicago. It has four courses tuned gG-cCc-eEe-AA, so uses the same chord shapes and scale patterns as the uke.
The banjolele is also tuned the same way as a ukulele.

You are right in thinking that to play simple song accompaniments on a uke is pretty easy, but, playing at a higher level, like Jake or James or folks like Marcy Marxer, Gerald Ross, Del Ray or Stu Fuchs, will take a lot of effort and practise.

I have changed tuning on one of my ukes to GCEG or open G in order to play some slide, but this is not a common practice.