Most of the time, you only need to use your forefinger and middlefinger - for the simplest phrases, even just one of them can suffice.
One of the ways to get a clean sound is to have your finger pluck the string, then keep going and land on the next string. For example, if you play an open G, have the finger land on the D string. This will help keep your finger in a consistent range (giving you a point of reference, which is always good), and will also mute the adjacent string, making it harder to have funny sounds coming from those strings. I usually keep my thumb anchored on the low E string, but on the top of the pickup (as in the picture) is equally valid. When I have my thumb anchored on the low E and want to play on that string, I move it to the top of the pickup. I find that easier, because I like the angle that it puts my wrist at.
If you play with two fingers, it's imperative to practice alternating between them. Otherwise, you're losing the speed that you can get from using two fingers.
For triplets - I like playing 3-2-1 (ring-middle-index).