Hi there!
The JR Pass only covers JR lines, and so Kintetsu trains are not covered.
For Kyoto, Osaka and Nara you will be able to get between these cities, but within the cities you will need to purchase tickets for the Metro or Bus as needed.
For Kyoto you can purchase a bus or metro pass once you arrive at the station (see the Tourist information at the station for more on this). For Nara you will need to purchase bus tickets as needed. Osaka has JR lines within it that you can use, but you will need to purchase metro tickets for using the metro.
As an alternative to tickets, you should also take a look at the Kansai Thru Pass, which covers virtually all non-JR lines in the Kansai area (Osaka, Kyoto and Nara are covered), including all buses and metro services. Please note that you'll really need to do a lot of travel to make the most of this pass, so in general I recommend purchasing tickets as needed.
Hope this helps!