When strumming, you use your thumb or finger to move your hand across the strings to play multiple notes. With fingerpicking, your hand stays in a steady position over the strings, and your thumb and fingers play individual notes. It's really that simple! You can "pluck" one string at a time with your thumb or a finger, or play multiple strings together with thumb and fingers, using a "pinching" motion, or play two strings at the same time with two fingers ("double stops").
It doesn't have to be fancy picking patterns at first; just try slowly playing the open strings one at a time, forwards and backwards: G - C - E - A, then A - E - C - G, and repeat.
Then slowly play a C Major scale up and down, starting with the open 3rd string: C - D - E - F - G - A - B - C, then C - B - A - G - F - E - D - C, and repeat. (Hint: that's frets 0 & 2 on the 3rd string, 0 - 1 - 3 on the 2nd string, and 0 - 2 - 3 on the 1st string & reversed.)
Keep doing those until they start to feel comfortable, and gradually speed them up a little, then try picking out single note melodies for simple songs like 'Frere Jacques' (Are You Sleeping), 'Row Row Row Your Boat', 'Mary Had a Little Lamb' & 'Polly Wolly Doodle'. Boom, you're fingerpicking!
Next, you can practice simple picking patterns on the open strings like: G - A - C - E, or C - A - G - E, or any repeating patterns, as exercises to feel more comfortable picking individual notes. Another good thing to practice is forward rolls, i.e. picking Thumb, Index, Middle (T-I-M-T-I-M-T-I-M...) and backward rolls, Middle, Index, Thumb (M-I-T-M-I-T-M-I-T...). Play them slowly and smoothly, over and over, until comfortable then gradually speed them up a bit. Then, play those patterns and rolls while holding chords, and it'll start to sound pretty good!
Lately, I've been practicing two-finger picking with alternating Thumb and Index finger (T-I-T-I-T-I-T-I-T-I-T-I-T-I-T-I...), and alternating Index and Middle fingers (I-M-I-M-I-M-I-M-I-M-I-M-I-M...), on one string and on adjacent strings.
The keys are: Keep your your picking hand relaxed and your wrist fairly straight. Keep your fingertips close to the strings and use very small finger motions to pluck the strings. Use whichever finger (or thumb) feels most comfortable to pluck each string.
Some people assign each finger to a particular string, i.e. thumb on 4th string, index finger on 3rd string, middle finger on 2nd string, and ring finger on 1st string, etc. Some people also brace their hand in position by keeping their pinky finger anchored on the top of the instrument.
I personally don't do either of those last two. It's limiting for me to always pick the same string with the same finger, and keeping a finger or two braced against the top creates a lot of tension in my picking hand and causes pain in my wrist and hand. On any given song, I just do whatever is the most comfortable and sounds the best to me, and I mostly use my thumb, index, & middle fingers. I tend to strum over the 12th-14th frets, but I fingerpick over the sound hole, because it sounds better there.
Hope some of this might be helpful!