Muting is not much used in soloing (but a jazz/ rock/ flamenco guitarist called Al Di meola springs to mind), lead runs generally are not muted but this technique is common in metal rhythm guitar.
Here's how: hold your pick between thumb and 1st finger and use the fleshy area at the bass of the thumb to mute the strings by placing it lightly on the strings near the bridge. This is easiest on the wound strings, less pressure is required on thinner strings. You should eventually be able to do this equally easily with up and down strokes.
If you play lots of fast things on single strings then you will never need muting runs with plenty of pull offs and hammer ons often leave the lowest note of the string to ring out.
If you find that strings ring out when you don't want them and you are not playing something where you need the open strings you can get a "hair bobble" like hippy dudes and women wear and stick it round the neck fairly low down. A bandana or random peice of clothing will also work but I don't recommend you make this a habit since it is limiting. I only do this if I'm playing a really difficult and very fast runs.
If you want to cross strings you will often find shapes that don't require you to remove fingers and replace them, all you need to do to mute these is release tension on fingers covering each string as you cross them. This takes practice.
For really fast runs you can cross strings and hammer on/pull off.
Good luck!