I am left handed, but I play righty, and I've always wondered why using your strongest hand on the fretboard isn't standard.
Now I know, it's because fingering the frets is almost exactly half the work! I've seen so many guitarists play beatifully, only to bash the strings with their picking hand, and not even know why they sound terrible.
Finger picking is not as daunting as many people think it is, and I will give you a very basic walkthrough.
There are two parts of your finger you use to pick, your fingernail, generally used to "wack" all the strings when playing a chord. This has a "hard" and edgy sound, compared to a mellow sound when playing with the flesh of your fingers. If you play chord progressions switching between flesh and nail, you can get some very interesting dynamics.
But how to pick individual strings? Most of the time, if you are hitting a single note, you'll use your thumb, index finger, or middle, and occasionally your ring finger. To pick two or three notes, you make a "pinch" motion, plucking two strings toward each other, and for most two note chords, you will use your thumb and index finger.
A technique used in almost all types of music, to either stop unwanted strings vibrating, or to create a subdued sound. You usually need a plectrum to play with palm muting, and it generally consists of simply resting your hand on the bridge, so that just some of the flesh of your hand is resting on all the strings above (the high E being the bottom string) the one you are playing on.
Where To Pick?
A lot of people pick in one static place, most do it between the neck and bridge pickup. However, a lot of different tones can be found by moving your pick along the neck. Right at the bridge gives a twangy treblysound, like a banjo, or maybe a sitar? Between the pickups gives a "normal" sound, concentrating on the mids. At the base of the neck gives a bassy sound. Even experiment with picking on the neck, around the twelth fret, this makes some chords sound very special!
To see what I mean, start with a G chord, and place your pick at the twelth fret of the low E, now pick in a Diagonal line, right to the bridge screw on the high E string, and you will get a cascade of different tones thrown at you. Beatiful!
Using a pick is the usual way to play electric guitar, and is so overlooked by guitarists.
To be quite frank, apart from in very specific situations, economy picking is the way to go. This means, you pick toward the string you're going to be playing on, and if you have more than one note on the same string, you use alternate picking (going up and down)
Other Right Hand Tricks
Remember, your tone, volume, and pickup selector are right at your fingertips! Use them! Flicking from neck to bridge pickup will give you that extra boost for a solo, and cutting the tone will give you a good smooth, bleeding tone for when you want to make those blues squeal.
And, remember that when you're tapping, you can use your right hand to give you five extra fingers to really shake the hell out of the fretboard
Have fun with your new hand!