Stanchinsky's Song Without Words is a simple but beautiful one. #2 is my favourite.
I wasn't contributing to this thread since I figured Bach inventions would be the first to be mentioned, but I guess not so let me do it. Bach inventions...Seriously though, these pieces aren't too demanding but have a really nice charm to them. The 2 part and 3 part inventions are great depending on your level. They're also nice for analysis and teaching voice leading principles.
I guess the OP's suggestions made me lean much more in favor of stuff from beyond the 18th century .However, and as much as I've always enjoyed playing Johann Sebastian's "Inventions" (they are hot stuff, to be sure), I rather agree with lelle on the fact that they can be a bit trickier than appears at first glance, and perhaps a tad above the technical level of what the OP was asking for. But if that's not the case, I guess the first, third and fifth of the two-part ones could be good choices.
One of my favorite "on the side" pieces is Debussy's The Snow is Dancing. It is a beautiful piece and it is not that difficult to learn.