The order of sharps is as follows:
F C G D A E B

The order of flats is the opposite:
B E A D G C F

That basically means that if A is sharp. D,G,C & F must also be sharp. If A is flat E and B must be flat.

To determine the key when using sharps find the last sharp and go up one half step. If the last sharp is C# then the key is D. If the last sharp is an E# then the key is F#.

For flats the key is the second to last flat note in the order of flats. If the last flat is Db then the key is Ab. When only B is flat you'll just have to remember that it's in the key of F.

I hope this helps. If not, I'm sure someone else can put it better. Also this only applies to major scales, but thats all you really need to know for now I assume.

First, I'll tell you how I identify the major key assicuated with a key signature. Next, show how to determine if the song is actually in the
relative minor key.

I identify keys a little differently than my teachers taught me (those who know me won't be surprised by this!). I've memorized the cycle of 4ths and 5ths, as follows:

C G D A E B F# going sharp, and
C F Bb Eb Ab Db Gb going flat

I'll number these:

C G D A E B F#
0 1 2 3 4 5 6 sharps

C F Bb Eb Ab Db Gb
0 1 2 3 4 5 6 flats

So, if I see 3 sharps, I'll run through my list of sharps: G D A. 3 sharps is the key of A major.

If I see 2 flats, I think: F Bb. 2 flats is the key of Bb.

Whether you are in a major key or its relative minor can usually be determined by looking at the final (resolution) node of the melody. If the key signature is 0 sharps or flats, and the final note of the melody is a C, than the key is C major. If the final note of the melody is an A, than the key is A minor.

Here's another example. Suppose the key signature is 2 flats. If the melody ends on Bb, then the key is Bb major. If the melody ends on G, then the key is G minor.

If you're wondering what the "relative minor" is, it's the minor key of the 6th note of the major scale, with the tonic counting as the 1st (not 0th) note. Here's the scale of C major:

C D E F G A B
1 2 3 4 5 6 7

The 6th note is "A", so the relative minor is "A minor".

Another way to determine the relative minor is to go three keys sharper on the cycle of 4ths and 5ths (wrapping around the cycle appropriately).

What if you have a key signature, say 2 sharps, but the last note of the melody isn't a D (key of D major) or a B (key of B minor)? In some cases, this is simply a melody that doesn't resolve. You might hear this in movie music, where a melody line is interrupted short of resolution to add tension to the moment. On the other hand, you might have what is called a "modal" melody, which means a mode (key) other than the conventional major or minor keys. (By the way, some music purists may object to the way I'm bluring the distinction between "mode" and "key". Tough.) For example, if there are no sharps or flats, and the melody ends on a D, then the melody may be written in the mode of D dorian. The modes (keys) other than the normal major and minor keys are common in European folk music, such as, say, celtic music.