Piano Forum

The first is a symbol I'm not familiar with, it makes it's first appearance on the first beat in the first bar. I'm thinking it's some sort of rest.

Are you talking about the upper staff? If so, does the symbol in question look like a "}". If so, it's a quarter rest.

Next, on the second beat in the first bar, there's a chord, BDF, and above it is a rest. What's this rest doing there?

Many scores don't have that rest. If it's a black bar, it's a half note rest. Generally, if you have notes with rests above or below them, it probably means that there is a second voice that happens to have a rest at that point, which is exactly the case in this piece.

What key is this piece in? C# and F# would be D Major, would it not? However, to my chagrin, I'm not familiar with all of my minor scales, and I would think that this piece would be in a minor key.

It could also be the relative minor (B minor). However, I believe it is actually D major.

Finally, and out of curiosity, what exactly is a Gymnopedie?

When Satie was introduced to the famous 'Chat Noir' cabaret in 1887, he did not want to be classified as a musician, so he declared himself to be a Gymnopediste, though nobody was sure what it meant. It is thought he got the name from a poem by Latour, which mentions 'gymnopedia' - describing them as naked Spartan dancing boys. He relished the confusion that his title caused, and used it as the title for three piano pieces for this reason. These dances are ceremonial choral dances performed at ancient Greek festivals. I'll refer you to the following two articles about this topic:

https://en.wikipedia.org/wiki/Gymnop%e9die
https://en.wikiquote.org/wiki/Gymnopaedia
 
(I googled all this within 2 minutes...)