Hi there,
You don't have to purchase a JR Pass, and can travel on normal JR tickets if you would like. The Japan Rail Pass offers unlimited travel on all JR lines for a 7 day period, but for the 7 day version, costs ¥28,300. This means you have to spend more than this on JR tickets in order to make one cost effective. In general, a return from Osaka - Tokyo is around ¥28,600, and so you will reach the cost of the 7 JR Pass in just 1 return bullet train trip.
To illustrate, lets take a look at some costs for the main stations in the basic itinerary I listed above:
1. Kyoto > Nara: ¥690, (45mins, 41km)
2. Nara > Osaka: ¥780, (55mins, 48km)
3. Osaka > Hiroshima: ¥9750, (104mins, 345km)
4. Hiroshima > Miyajimaguchi: ¥400, (26mins, 21km)
5. Miyajimaguchi > Tokyo: ¥17850, (331mins, 916km)
6. Tokyo > Nikko: ¥5230, (121mins, 150km)
7. Nikko > Kansai airport: ¥19190, (367mins, 763km)
TOTAL: ¥53890 for 7 legs
As you can see, the cost of purchasing single JR tickets (¥53,890) is far higher than the ¥28,300 cost of the 7 day JR Pass, and so in this case it would make sense to purchase one. If you don't think you'll do this amount of travel, then purchasing single tickets may make more sense, however it really depends on the exact amount of travel you plan on doing.
Hope this helps!