The 7 Day JR Pass will allow you to ride the Shinkansen (the bullet train).
For the route from Tokyo to Kyoto there are three services of Shinkansen
1. Nozomi
2. Hikari
3. Kodama
Nozomi is the most frequent but you cannot use the JR pass with the Nozomi Shinkansen.
You can however use the Hikari Shinkansen, there are about two Hikaris per hour between Tokyo and Kyoto, you can make free reservations with your JR pass. You can pull up schedules using
http://www.hyperdia.com (in advanced options UNCHECK NOZOMI). Ignore all the fares, you don't pay them with the JR Pass.
Kyoto - Nara is rapid trains no reservations, just show your JR pass when using JR trains.
Kyoto-Osaka, you can use JR Special Rapid trains on this section, again just show your JR Pass. You can also use the Hikari or Kodama Shinkansen between Kyoto and ShinOsaka Station, the trip is so short between Kyoto and ShinOsaka you could probably just opt for the unreserved section of the train (the shinkansen has both RESERVED and UNRESERVED) sections.