For a start learn the Carter family lick with thumb and index finger, including playing the melody with your thumb on breaks. Also learn a few 4 and 6 beat arpeggio patterns (such as Thumb, index, third/fourth, index). With those you'll be able to cover a lot of what she did early on. Beyond that you'll need start messing with some slightly more complex finger picking. A lot of what Joan did was to just do simple things really well, so don't be in a rush to get too fancy too fast.