First of all, we will be leaving our picks behind. This first lesson will concentrate, on a simple exercise using only the 1st and 3rd (or 4th depending on how well you can stretch) fingers of your fretting hand and the pointer (1st) finger of your usual picking hand. If you wish to practice using the middle finger of your picking hand you may, this would enable you to continue holding your pick with your thumb and 1st finger, which is helpful while playing live.

This first exercise has become somewhat of a cliché over the years. Technically, it's the notes in an Amin triad (A-C-E) played in an ascending pattern.

( )= tap fret

E|---5h8(12)---5h8(12)---5h8(12)---5h8(12)-----------------------|
B|---------------------------------------------------------------|
G|---------------------------------------------------------------|
D|---------------------------------------------------------------|
A|---------------------------------------------------------------|
E|---------------------------------------------------------------|

Let's talk about the technique itself, and how you would go about playing that piece.

The 1st finger of your fretting hand should be on the 5th fret, and there's really no reason for it to move from there, keep it there.
Now to get things started, you need to use your tapping finger to sort of pick the string, we're not tapping yet, just pick the string with your tapping finger so the 5th fret sounds.
Now, with your 3rd/4th finger, hammer on to the 8th fret and NOW give the 12th fret a firm tap with your tapping finger! Keep your finger there on the 12th fret because here is where it gets tricky.
Immediately after you tap the 12th fret, you need to take the 3rd/4th finger of your fretting hand OFF the 8th fret.
Now with your tapping finger, make a scooping motion towards the other 5 strings, this will pick the string and make the 5th fret sound, and viola! We are back where we started.

Practice your technique for awhile, then try it on other strings!

E|-----------------------------------------------------5h8(12)---|
B|-------------------------------------------5h8(12)-------------|
G|---------------------------------5h8(12)-----------------------|
D|-----------------------5h8(12)---------------------------------|
A|-------------5h8(12)-------------------------------------------|
E|---5h8(12)-----------------------------------------------------|

That's not meant to be played the way I wrote it, just practice the tapping technique on each string.