It all depends on the context. Sometimes, you want to let the other strings ring, sometimes you don't.

Let's say you don't, notes that are fretted (not an open string) can be muted by releasing the pressure on the fretboard, just like you said. If the note is an open string, then you can mute the string with either your left (fretting hand) or right hand (picking hand).

Let's say you want to play the high e string (open 1st string) then play the 2nd fret 2nd string. If you don't want the high e string to ring when you pick the next note, you can angle your fretting finger a little so that it touches the 1st string, thus cancelling the ringing. You could also use your right hand to mute the string using your finger. So you would touch the 1st string with your picking hand fingers as you move to the 2nd string.

Hope this helps.