The Key:
First of all, remember what a time signature is.  In 4/4 time, for example, there are four beats to a measure and each quarter note gets one
beat.  That's the key to counting numbers.

So, with this in mind, we know the following:
 1) There will be four beats in each measure (i.e. you must use
counting numbers 1,2,3,4).  If you only count 3 of those, it's wrong.  If
you count 5 (i.e. 1,2,3,4,5), it's wrong.

 2) The numbers will suffice if the song is in 4/4 and contains
only quarter notes or higher.  For example, if a song has a whole note, it
takes up a whole bar, and under that whole note you'd place 1.  You'd
count the 2,3, and the 4 to yourself.

 3) If the song is in 4/4 and some of the notes are eighth notes,
you need to add "divisions" to the numbers.  For this, you would use "and"
or  the ampersand "&" in addition to the numbers, so the counting numbers
are now  1 & 2 & 3 & 4 &.  If a song has a measure with 8 eighth notes,
you'd use 1 & 2 & 3 & 4 &.

If the measure has a half note and four eighth notes, you would use 1 3 & 4 &,
to indicate that the first note gets played on the count of 1 and the four
eighth notes get played on 3, on &, on 4, and on &.  The 2 is not
included, even though you count it to yourself while performing the song,
because it gets counted while you're holding the half note.

 4) If the song is in 4/4 and some of the notes are sixteenth
notes, you need to add "subdivisions" to the numbers.  For this you would
use "e" and "a", so that the beat subdivides into 1 e & a 2 e & a 3 e &
a 4 e & a.  In this case, a measure with a half note, four sixteenth
notes, and 2 eighth notes would be counted 1 3 e & a 4 &.

 5)  In all cases, treat rests the same as notes, except indicate
rests with parentheses.  So, I measure with a quarter note, a quarter
rest, and a half note would be counted 1 (2) 3.

Some examples:

4/4 h h   | q q q q    | dh      q     | q  h      q      | q q dq     e      | w             | qr   h      qr ||
       1 3  | 1 2 3 4    | 1        4     | 1 2      4       | 1 2 3        &     |1              | (1)2      (4) ||

3/4 h  q | q e e q | dh       | qr  q  qr    | q h    | q    er  e   q ||
      1  3 |  1 2 & 3|  1        | (1) 2  (3)  | 1 2    | 1   (2) &   3  ||

2/2  h h    | w    | h  q q     | hr    h  | q q q q     | w          | dh       q     | hr     q q ||
       1 2    |1      | 1  2 &    |(1)  2   | 1 & 2 &    |  1          |  (1)     &     |(1)    2 & ||

4/4  q  e e h      |  s s s s e  e  dq      s s      | h        s s e   e s s    | s s s s e  e  qr    q  ||
        1 2 & 3      |  1 e &a 2  &  3        &a    | 1        3e&    4 &a   | 1 e& a 2  &  (3)  & ||