For speed picking, most electric guitarists use something called alternate picking, or variants of it. Learning it early would be a benefit for your hand/finger coordination. There are a lot of good easy-to-find free lessons online, i recommend Pebber Brown's videos on YouTube (but staystay away from his left hand videos, they don't explain good hand posture enough) but I'll provide a method below.
For the pick to not "catch" on the strings, angle it downwards (upwards is sometimes useful but not as often), meaning it shouldn't be parallel to the string when hitting it. Grip the pick between your thumb and the side of the outermost joint of your first (index) finger. Point the tip perpendicular towards the strings. As for your other fingers, you can use them as a pivot point on the body of the guitar, if you find it very hard to hit the right string. Otherwise curl them up into something close to a fist, and use your forearm and/or palm for pivots.
The movement needed can be created in five (5) ways at least. The most common one is using wrist motion (needs clarification?). As you move your wrist, the force should come mainly from a twisting forearm motion. The next three are finger motion, forearm motion and shoulder motion, out of which shoulder motion is probably useless. Finger motion consists of flexing the thumb and first (index) finger joints so that the pick moves in the direction of the edges. As for the other two you can probably guess. Now the first three mentioned have their respective uses, but if you strive for efficiency, you want to use two or more of them. If a muscle group only has to perform part of the motion, it is under less load and can perform better. So when you are confident in the basics, combine two types of motion (finger and wrist recommended) and reach for speed!
[will addAfter you are confident in alternate picking, it's time to learn economy picking. This is a section herereal virtuoso technique, and is more efficient for difficult string-crossing stuff. The basic rule is, instead of alternating up/downstrokes, you always pick towards the string where your next note is. For example: even if your previous stroke was a downstroke on economy picking later todaystring 4, don'tyou use a downstroke to move to string 3. Same techniques apply, you just have time right now]to think differently about it.
The rest is mainly diagnostic. For example, if you find your down strokes are louder than your up strokes, practice up strokes and try to normalize the volume between strokes. Articulation exercises are good for this, they make you listen to yourself, try emphasizing every third note so that it is not in sync with up/down strokes.
And finally, some additional considerations:
Using a thicker, stiffer pick with a sharper point can make speed picking easier, though strumming with such a pick requires a much more relaxed wrist.
Avoid tensing up in your forearm and wrist. Aim for a relaxed picking style.
A (competent) personal guitar teacher will be able to look at your technique and help you improve. If you happen to be living in a country where education is available (meaning affordable), what are you waiting for?