1. I want to travel from Narita Airport to Osaka Castle for a tour. 
What is the most direct route (Least transfers, least modes of
transportation) by bullet train?
2.  From Osaka castle, how do I get to Kansai Airport?

First question:
1. From Narita Airport take the JR Tokyu Narita Express to Tokyo Station.
2. Take the Shinkansen (Bullet Train) Nozomi to Shin-Osaka Station
(about 2 1/2 hrs).
3. Take the JR Tokaido Sanyo Main Line one stop to Osaka Station.
4. Take the JR Osaka Loop Line (outer loop) 4 stops to Osaka-Jo Koen
(Osaka Castle Park station).

Second question (even easier):
1. Go back to Osaka-Jo Koen station and get on the same train -- JR
Osaka Loop Line (outer loop) to Tennoji Station (5 stations).
2. Transfer to the JR Tokyu Haruka directly to Kansai Airport (about 30 mins).

Hope this all helps!