Here's a great fingerpicking pattern to get started with. Try this pattern while fretting an Eminor chord. Notice that the thumb (P) alternates on the Lowest 3 strings (bass strings). The index finger (I) picks the G string, middle finger (M) picks the B string, and the ring finger(A) picks the high E string.
A very beautiful pattern when picked slowly, and distinctly Spanish when sped up. Try to arch your wrist slightly and keep your fingers curved as if you're holding a ball.