The quickest route from Kyoto to Shin-Osaka would be the JR Haruka Airport Express, since you can't use the Tokaido shinkansen. You can't reserve it for that leg but just walk on and find a seat in the non-reserved car. The Haruka is rarely full outbound from Kyoto in the morning. Afternoons are busier. There is also a local train but it will be slower. From Shin-Osaka to Okayama you can use the shinkansen but also only in a non-reserved car.

I think that last leg will be pretty simple if a little expensive since you will be on the shinkansen. If you have time, you can just buy that ticket at Kyoto before boarding whichever train you will use to Shin-Osaka. You could also buy it at Shin-Osaka or Okayama, whatever is easy. Just make note of what time your preferred train will arrive at Okayama and leave some time to make the connection. Hyperdia will help there. I think I would just exit the platforms at Okayama, turn around and re-enter on your paper ticket and find the right platform. Depending on what time of day it is and how much time you have, Okayama might be a good place to grab a bite to eat or a lunch box to take with you before going back in, since there are many services.

If that all seems a bit complicated, just exit the platforms at Okayama, find the ticket window (midori-no-madoguchi) and buy a round trip ticket to/from Hiroshima. Okayama is a major rail hub and there are trains leaving every few minutes. Since you are paying for this part of the trip, you can reserve seats.