Takegawara Onsen,one of Beppu's oldest, is a must see. It offers two types of baths -- either a simple, scaldingly hot uchiburo (be sure to bring your own soap and wash towel, as the 100 yen doesn't include any extras!) or the more unusual sand baths (which cost 1000 yen). As with many of the simpler baths, this is one where you wash by scooping water out of the bath itself and rinsing well before getting in. It's hot, atmospheric, and relaxing, surely one of the most memorable places you'll stop in during your Kyuushuu trip.
Beppu's main train station is centrally located and offers easy access to hotels, hostels, and plenty of baths. Walking towards the water will take you to the bar district, where along with plenty of bars and clubs you'll find shopping arcades and restaurants. Nightlife here is huge, some of it seedy. But the variety of baths, the quality of the water, and the history here make Beppu a prime spot to stop.
The youth hostel in Beppu is great for budget travelers -- the owner is a great source of info for onsenning suggestions, and he even serves beers and local kabosu (a type of citrus fruit) liquor in the hostel's own bar. The hostel itself has a nice onsen too, which makes for relaxing return after a long day. The down side is that lights out is pretty early.
People will a bigger budget will find good value in the hotels close to the station. Many of them have onsen baths as well, and you can ask for sudomari plans which strip down the cost to a night's stay only, without any meals.