- Peace Museum (if you don't cry I'd be confused)
- Miyajima (go on a weekend and some University students will probably give you a free tour to practice their English)
- Hiroshima Okonomiyaki (mo something?) - fucking AWESOME
- Great vibe
JM definitely clarified what's not amazing about Osaka: it's like mini, boring Tokyo. It has little to define it as a place (other than lukewarm versions of Tokyo's shopping and energy) and little really interesting history (from my mind). I remember being on a ferris wheel, looking out over 100s of kilometres of Osaka and thinking "well, this place is essentially warehouses".
+1 on the Nara recommendation - can be 1/2 a day or a 1 day, but Todaijii (might have mis-translated) is gorgeous, and the deer are funny.
I would also heartily recommend finding some time for a private onsen (hot spring), especially near a mountain/town. Relaxing, intense, oddly violent, but totally Japanese and totally unforgettable (water that hot isn't easy to forget!)