It depends on the style you want to play. Fingerstyle could range from Mark Knopfler to Joe Pass to Paco de Lucia to Extreme. The more you practice, the more comfortable you'll be at it. How long is relative because some people learn faster than others. What sort of music do you want to play?
You don't need nails on either but it's good to have them because they give you a better tone (even on a steel string). Like suvro said, the nylon string guitars are more comfortable to play fingerstyle classical on because of the wider spacing between strings but I wouldn't call it a must. I think you can go to anyone who knows how to play the style you want to learn. Just hope that the people teaching you are sound guitarists and good teachers.
And surely you’ll buy your pint cup !
and surely I’ll buy mine !
And we'll take a cup o’ kindness yet,
for auld lang syne.