Let's introduce a mixed example.
The quarter note is obviously beat 1 because from the time sig you know there are 4 quarter notes per measure. You also already know one half note = 2 quarter notes therefore the half note must be beats 2 and 3. Finally, you know that two eighth notes = 1 quarter note so they must be the "4 +".
When many different kinds of notes are intermingled, it starts to become tricky to count. Musicians will sometimes subdivide the notes so the counting flows more easily. Let's use the above example, but this time sub divide it.
Here every note in the measure is subdivided into 8th notes thus making it a lot more "fluid" to count. Its pretty easy to understand too... one quarter note is two 8th notes, so it gets "1 +". The half note is really four eighth notes so it get "2 + 3 +". And the each 8th note get a half so one is "4" and the other is the "and" of 4.
Here would also be a good place to throw in a few examples with rests. These will just show the counting and will not explain them. Just think of the rests in terms of their corresponding notes and you'll have no problem!
Counting the 16th note.
Basically counting 16th notes is similar to 8th notes except that you need to add more things to count with. I was taught using "e" and "a", but feel free to use what you want. Each part, the "1", "e", "+", "a" are all 1/4 of 1 quarter note. Together they add up to 1 beat according to the time sig. (4 sixteenths = 1 quarter)
Different time sigs and different notes.
Here you are.. the top of the note hill. Just look at these and the counting section is over!
Remember.. from this time sig you are counting the 8th notes.
Remember you are counting half notes, and therefore you have to subdivide the eighth notes and quarter notes accordingly.