Flying into Hiroshima in Dec for 8 days

We will be flying into Hiroshima in early to mid Dec and have 8 days to do as we please..
How much time should we spend around there for the most important tourist sights? What kind of timing is practical to train to Hakone/ Nara and back?
Any point in visiting the parks in Nara during Dec?
Any other itinerary suggestions, please?
Will hotel/ ryokan rates be very high during this season? We are looking at about 150usd/ night.
Thank you!

Depends on what you wanna see. But Hakone isn't anywhere near Hiroshima.

Nara is 3 hours away, and Hakone is 4 hours away.

I'd look at a map and decide where you wanna go. With only eight days, I'd suggest just Hiroshima and Osaka/Kyoto region.

Two days is a good amount of time to cover Hiroshima City and nearby Miyajima. For the city, aside from the Peace Park, the Shukkeien Garden is very nice. Plus don't miss the beautiful yet nearly unknown Mitakidera Temple. Sadly most people do - but it's so peaceful and enjoyable without the crowds.
https://goo.gl/fAz4ax

You could also see Iwakuni just past Miyajima, with the loveliest old bridge in Japan.
https://youtu.be/0COOZXi66mM

The other big gem in the area is Himeji, with the famous castle and the gorgeous Kokoen Garden nearby.
https://goo.gl/65ZtRi

Hakone is almost up to Tokyo - not realistic. Perhaps you mean Hikone? Nara is well worth your attention. Nara Park has a lot of the best, and the Todaiji Temple is a must.
http://bit.ly/1Rhxg0E
It's about 2¾ hours away by train with a good connection.

For something a bit different, maybe think about a ferry from Hiroshima over to Shikoku...

There's Kure, with the JMSDF museum and the Yamato Museum. In Onomichi you'll find a quaint little picturesque town on a hill. And Saijo has many sake breweries.

The following pass may be useful for you, covering both Hiroshima and Nara:
https://www.japan-guide.com/e/e2361_kansai_hiroshima.html

As you can see from the pass that it covers a wide area, so I am sure that you find some cities and attractions that will catch your interest.

A list of other passes which is useful for Hiroshima: https://www.japan-guide.com/e/e2357.html , scroll down to Chugoku.

Personally I enjoyed visiting Matsue. Discount tickets for the highway bus:
https://www.visit-matsue.com/_assets/page_files/files/000/012/320/2621...