Hi Teresa,
You won't have to worry about what trains to use or how to travel. This because the JR Staff is here to help. When you are in Japan simply tell the JR Staff where and when you wish to travel, show your JR Pass and they will find the best route for you and print the tickets without you having to pay anything!
I took a look at all your routes and they are all pretty straightforward and I don't think that you will have to worry at all.
I've added notes to every route to make the travel as easy as possible.
Narita Airport - Tokyo. From Narita Airport take the Narita Express to Tokyo, this is the best way to travel there and will bring you to Tokyo directly.
Tokyo - Disneyland/Yokohama/Kamakura - Tokyo
All these places are covered by the JR Pass and can be reached directly from Tokyo there's actually a lot of trains going to each direction and trains leave about every 10 minutes so just go to the station ask the JR Staff what train is leaving and enjoy the ride. The station for Disney land is called maihama and can also be reached directly from Tokyo
Tokyo - Nagoya
There's a direct Shinkansen service between Nagoya and Tokyo, leaving at least 4 times an hour. So head to Tokyo station and travel board the Shinkansen, don't forget to reserve your seats free of charge at any manned JR counter before leaving.
Nagoya - Takayama
This is also a direct route with on the Hida Wideview train and is one of the most beautiful routes in Japan. Trains leave every 30 minutes.
Takayama - Kanazawa
This may be the only route that is a little tricky as train connection and transfers vary and trains ride somewhat irregular. I would recommend trying to catch this train connection leaving Takayama at 11 a.m. as it is by far the most convenient way to travel. This route like all others is covered by the JR Pass.
Kanazawa - Kyoto
This is a direct train connection departing 1-2 times an hour and is a very nice ride. The train is called the Thunderbird Express, which is think is pretty cool!
Kyoto - Nara - Kyoto
Nara is easy to reach from Kyoto, just take the Nara line directly from Kyoto station leaving multiple times an hour and will reach Nara in about an hour.
Kyoto - Kobe
Like Nara, Kobe can be reached directly from Kyoto, the train connection is called the ShinKaisoku (new rapid service) departs every 10 minutes and stops in the center of Kobe. The central station in Kobe is called Sannomiya and you may want to get out here depending on your accommodation.
Kobe - Hiroshima
In Kobe travel to the Shinkansen station called Shin-Kobe you can take the Sakura Shinkansen directly from there to Hiroshima. Also leaving multiple times per hour.
Hiroshima - Miyajima
Take a local train to Miyajimaguchi, here you can board to ferry to Miyajima island, both the train and ferry are covered by the JR Pass.
Miyajima - Osaka
Take the Ferry and Local train back to Hiroshima, you can board the Bullet train (shinkansen) to Shin-Osaka where you can take a local train into the city depending where your accommodation is.
Osaka - Kansai Airport
There's a direct train service from Osaka called the Airport Express going to Kansai Airport (KIX) leaving twice an hour.
All trains are covered by the JR Pass!
I hope this helps!
Daniel