There are lots of electric guitar players picking without a pick, e.g. Mark Knopfler. Jeff Beck also does some creative finger picking on his electric guitar. Some folks do both at the same time, i.e. using a pick and fingers together (and that's not a thumb pick). Do whatever works best for you!
Traditional acoustic-guitar fingerpicking is done with the thumb managing strings 4-5-6 and the first three fingers managing strings 3-2-1, but this can be varied a lot. It works for electric guitar too, but on electric you'll often want to do a lot more of muting so there are some other tricks to be aware of - and I'm not an expert on electric guitar - I tend to play it as I play an acoustic.
Edit:
On the other hand, learning to use a pick isn't too difficult. The initial problem tends to be to lose the grip and thus lose the pick.. on electric it'll just fall on the floor, but on acoustic it always falls into the soundhole and you'll see the guitarist frantically trying to shake the pick out! The problem goes away after a while though, I used to lose the pick into the soundhole all the time in the past, but that hasn't happened to me the last fifteen years or so.
When you use a pick on guitar you should start with sweep-picking, that is when playing notes in sequence you move the pick first down, then up, then down, and so on. I.e. first note: Pick down. Second note: Pick up. Third note: Pick down. This will lead to smooth, easy playing (and sounding). First learn the rule (=always sweep-pick). When it's second nature you can start breaking the rule, i.e. use all down-strokes or up-strokes selectively if it fits the music better.
(Edit: The way I used the term "sweep-picking" may instead be known as "alternate picking". "sweep-picking" is sometimes used for a technique where you still alternate, but you play the pick through several strings before you change direction. You don't strum, you still play individual notes, but you can "line up" the notes of, say, a melody, on different strings so that you by playing a down-stroke over some or all of the strings get a little melody. Then you could use an up-stroke the same way to continue the melody. It takes some planning though.. just alternating down-up-down-up over individual strings (moving to other strings as necessary while keeping up the down-up-down-up) is reasonably easy to get the hang of.)