Legato is simply a matter of fretting a new note while the string is still vibrating from the last note. This process gets helped along by some timely hammer-on and pull-off action. You may need to use your pick each time you move from one string to the next one, but the hammer'n'pull technique should cover most of what you do.

For practice, fret a note with your index finger and hammer'n'pull the note one fret up with your middle finger. When you can keep that trill going for as long as you want to, hammer-on the next note up with your ring finger, pulling off to the index finger, pulling off to your index, and then hammering your way back up again. Then add the next note up, played with your pinky. When you've got that down, you can add jumps to other strings.

As with any new technique, you have to start with a small step, and progress further when you can play that step accurately. Forget about being fast. By the time you're good you'll also be fast.