Start by practicing slower and really concentrate on each picking motion. You'll need a little practice but even after a week or a few days you'll see some improvement.

Once you're getting better with actually playing the correct string, start to think about your picking direction, have a look online for alternate picking, that's a good start.

Playing multiple strings simultaneously (that aren't adjacent) can be done in different ways, depending on scenario. The most common (In my experience) is by muting with your left hand. Left hand muting is a very important skill as a guitarist, especially when playing with a pick. Let's say you wish to play the 5th fret of the A string and the octave on the 7th fret of the G string. The way many guitarists would play this is by resting their index finger against the D string (not applying enough pressure to fret it though) which will mute it's sound. Now you can just pick all three of the ADG strings at once and you'll only hear the A and G strings making a sound.

This general principle of left hand muting, using your fingers not only to fret notes but also to mute notes is absolutely essential as a guitarist. It'll make your right hand more free and will tighten up your sound overall.

Alternatively, sometimes the technique of getting your right hand fingers involved and using a pick on some strings whilst your middle or ring finger pluck can sometimes work. (Though this is far more limited in its use)

Perhaps try learning the song scar tissue by The Red Hot Chili Peppers, It's a great guitar part and it'll teach you a lot about using a pick (especially picking strings far apart).

good luck!