Most of musical notation is about fractions. You take each bar, and divide it up. In 4/4 time, a semibreve () lasts a whole bar, a minim () lasts half a bar, a crotchet () a quarter, a quaver () an eighth, and a semiquaver () a sixteenth. Anything with a dot to the right of it lasts half of its length again. Anything with a line joining it to the next note means it's the same note held down longer.
So if you'd like to write a note that lasts three quarters of a bar, you'd draw a minim with a dot to the right of it.
Rests are the spaces between the notes, so you know where the gaps go. They look different, but they have the same names and deal with the same fractions. There's the semibreve (), minim (), crotchet (), quaver () and semiquaver().
The time signature tells you how long each bar lasts. 4/4 time means it lasts four crotchets, so all your notes had better add up to the same length as four crotchets per bar. 3/4 time means it lasts three crotchets, so all your notes should add up to three crotchets per bar.
So now you know when the notes start and finish, and when there are gaps between them, you just need to know the pitches. (It's slightly different for drums, but I'll gloss over that for now.)
The swirly G thing () is a treble clef. The line inside its florid loop is the G above middle C. The funny looking F thing () is a bass clef, and the line between its two dots is the F below middle C. Each line and each gap between the lines is for one letter of the alphabet, C, D, E, F, G, A, and B, looping through the octaves.
The keys of F♯ major and D♯ are a bit of a hack, as they have to crowbar F into being called E♯, but otherwise it's a pretty solid system.
So now all you need to know is which notes are sharp (♯) or flat (♭) or natural (♮). There are symbols to depict all of these written next to the notes, but to make life easier, the symbols are written at the very start of the stave (the horizontal lines), so you can just remember what's there and look along that line or gap and keep it always sharp or always flat, unless told otherwise. Anything with neither a sharp nor flat symbol at the very beginning is considered natural, as in neither. The combination of sharps or flats that always apply, of course, tells you which key the piece is in, so that's the key signature. For instance, if you see three flat symbols at the start, you don't *really* need to check which notes they're for as only the keys of E♭ and its corresponding minor key, C minor, have exactly three flats in them.
That should be enough to get you started. For anything more complex, buy a cheap second hand book on music theory and/or read documentation online.