I wasn't thrilled by Osaka but wow, I loved Kyoto. Apparently we are "temple people" and we are planning on going back in about 3yrs time.
The train between Osaka and Kyoto is either 40mins (slow standard train) or 12mins (shinkansen or bullet train), so there really isn't reason to change hotels unless you want to.
Sadly 6 nights isn't enough time but personally I would pick Hiroshima over Osaka but that's because Ive seen both.
I had exactly the opposite experience to you! LOVED Osaka (possibly my favourite city in Asia) but didn't like Kyoto.
Osaka
Big enough that there's plenty to do, small enough to be fairly easy to get around (compared with Tokyo).
Very few tourists I think because fewer foreign airlines fly into KIX vs NRT and most tourists don't venture 'all the way south'. It's a very surreal feeling to arrive into the main train station during rush hour, see literally thousands of people streaming past and realise you are the only non-Japanese people in the entire station!
Food capital of Japan - lots of really interesting food options.
I agree with the comment that its where to see the real Japan. I had local school girls asking to have their photo with me because blonde hair is so unusual there.
Osaka castle and aquarium were great. Again, big enough to be interesting, small enough to be manageable.
I think we had 3 nights there which was just right.
Kyoto
In comparison, it was overrun with tourists. They were everywhere.
Seemed quite spread out and more difficult to get around (many people rent bikes for this purpose).
Seemed much more expensive and a lot more tourist-orientated options (tacky souvenirs, western restaurants etc).
We joined the hordes of tourists trying to get a glimpse of a geisha in the backstreets one evening. Don't bother. The back-streets are so weirdly clean and new they feel like a movie set and the closest thing we saw to a geisha was american tourists who had payed hundreds of dollars to be made up to look like one (very, very tacky).
We had 2 nights here and I was pleased to leave.
Mickeymouse - 6 days to do Osaka, Kyoto, Hiroshima and Nagoya is going to be far, far too riushed. I would definitely drop Nagoya (I can't think of anything to do there) and choose between Hiroshima and Kyoto. Hiroshima is actually quite a charming little city - I'd lean more towards that. I would do 3 nights Osaka and 3 nights either Hiroshima or Kyoto if I was you. 6 days in Tokyo may also be a little rushed - I had 9 there and still didn't do everything I wanted to do!