What does the first note below mean? Does it need to be pressed down while playing the second notes? If not, why are there two stems, one pointing up, and one pointing down?
Dave is right, but there's a little more to it. You can break the part up into two different lines. One that looks like this:
And another that looks like this:
When you put them together, you get the two part represented by different stems. It's pretty much telling you to hold the first note for the length of a quarter note, but play the set of notes while the first is sustained.
There are two voices in this music:
f <a d> e <g c#>as eighth notes
This is the way to write music where more than one voice happen to execute the same note at the same time.
If you were to play this on a piano (as an example) the e and the f should sustain for the full quarter note. If the composer had intended them to stop the downward stem would have a flag to indicate that the lower voice only executes an eight note, and include a rest to fill out the beat.