Hello: We would like to see the Peace Memorial Park & Museum. About how long should we plan to see the sights, without rushing thru? Thank you!

For most people, a half day is sufficient. Different people react differently; but if you are deeply affected by what you see at the museum, you may want to spend a few hours afterwards wandering around, and you can spend that time well by looking at the various memorials in the park that surrounds the museum, or walking through the downtown area. There is also the former Bank of Japan building, within walking distance of the main museum, that includes a small museum space with additional exhibits.
My wife and I were short on time so only allowed one afternoon and evening for Hiroshima. He has lunch then spent about two and a half hours to go round the museum and then the rest of the afternoon in the park.
In the evening we then walked round the peace park again and went in to the side street to see the Hypo Centre being the actual point of the bomb.
While we did see all the main sights I wish we had allowed a full day as the whole place was very moving and there is plenty to see, in addition Hiroshima is the type of place that many of us will only ever visit once so do you really want to be rushed?
I would also suggest that you have a good look at the museum web sight as this can give you a very good idea of what you can see when you are there.