The JR Pass isn't all that convenient within Tokyo or Kyoto, either, but it does save you the hassle of buying tickets. For Kyoto, get a bus pass, sold in increments of 24 hours, since buses are the way to go if you are trying to hit the major sights.
Here's what you can use the JR Pass for:
Nagoya to Kyoto--any Hikari Shinkansen will take you there.
Within Kyoto, Kyoto Station to Nijo Castle, Arashiyama, Nara, and Osaka. Note that the Shinkansen route takes you to Shin-Osaka, which is kind of out of the way but is connected to the subway. If you take the Kansai Main Line, it is much slower, but you end up at Namba, which is more in the middle of things. Either way, before you go to Osaka, read up on it and figure out exactly what you want to see, since it is a large, sprawling city. The Kanjo Line, the JR-run commuter line, gives a good overview of the city, and sometimes it is elevated and runs so close to buildings that you can almost spy on the people inside.
Kyoto to Tokyo, definitely. You can't ride the Nozomi with a JR Pass, but there's only about half an hour difference in travel time between the Nozomi and the Hikari on this route. (The Hikari stops at more places.)
With a JR Pass, it is worth it to take the Narita Express to the airport, as long as your pass is valid on the last day. It's about an hour, but it's a comfortable, non-stop ride from Tokyo Station. If you don't have a JR Pass, take either the much cheaper Keisei Skyliner from Keisei Ueno Station (just a short walk from JR Ueno) or the Limousine Bus from the Tokyo City Air Terminal or from some hotels. The four- and five-star hotels tend to be served by the Limousine Bus. For the rest, you're better off with either the Keisei Skyliner or a taxi to the City Air Terminal.
(There are multiple ways to get almost anywhere in Tokyo and Osaka, which is convenient, but it can be confusing to a first-timer.)