About the mute-technic: You have to lift your finger slightly from the fretted note, just after you've picked it. Only lift it so it'll mute, not till it will bable on the open string. Also you can mute with your picking hand. On guitar, we use the palm, but on bass you'll use the picking fingers. Most bassplayers use two fingers alternately when picking, kind of the same when guitarist pick up and down. Altough the best thing to develop as a bassplayer is picking with 3 fingers ALTERNATELY! This no piece of cake to learn! But it'll surely pay off some day.

Oh damn, back to the question.
If you start by picking with your indexfinger on the low E-string, fretting an A (5th fret), obviously the next pick-finger is the middlefinger. JUST after you've picked the note, land the middlefinger on the string, DON'T PICK IT, and at the same time release the pressure on the fret (as I described before). This way you'll mute the hole string, above the pickups AND above the fretted note.
So after that, you'll fret another note, pick it with the middlefinger and again, JUST after, land on the string with the index again (or the ringfinger if you want to try with 3 pickingfingers), and again mute above the fret just by releasing the fretted note but still keeping the finger in touch with the string.

Phew! Hard to explain, even for a dane like me. Hope you can use it.

Fiddles
Trolle