I have been to Nagasaki and Hiroshima and i liked Hiroshima alot better. I have also been to both A-Bomb Museums and the one in Hiroshima is definetly worth it. The Peace park around the A-bomb dome is also very nice and i think you can take a boat trip to Mijayijima from there. I stayed in the J-Hoppers in Hiroshima wich was really nice because of the great staff and the good Location (very close to the Peace park).
If you want to stay at a hotel but it shouldn´t be to expensive than i can recommend
Hotel Reservations - Hotel Chain Toyoko Inn . This hotels are nearly everywhere, always to a decent price and easy to book online. Rooms are small but you get everything you need and the Hotel looks the same anywhere you go.
In Beppu you don´t have to stay at a place that is that expensive. I stayed at a place (a ryokan) with my brother that was recommended by the tourist information and we payed 12000 yen with breakfarst, got our own nice room, could reserve the rotemburo (outside onsen) for 1 hour and the staff was nice. The Shower, Toilet was not in the room though but shared.
In Koyasan i would recommend to stay at one of the Buddihst tempels in town. Its your best chance to see a morning prayer and the vegeterian food is really nice. Don´t stay at the cheapest one though, i have been there and it was ok but not nice. The Tourist information in town can help you, its right by the bus stop.