Arima Onsen is one of Japan's Three Old Hot Springs, with a recorded history going back to the Nihon Shoki, published in 631. The waters of Arima come in two flavors: the Gold Spring (金泉 Kinsen) is yellow-brown thanks to iron and salt, while the Silver Spring (銀泉 Ginsen) is colorless but mildly radioactive.
Technically, Arima Onsen is a part of Kobe, but it's on the other side of Mount Rokko and thus a separate city for all practical purposes.
The practical way is to take the Hanshin, Hankyu or Sanyo Line to Shinkaichi (新開地) just west of Kobe, then transfer to the Kobe Railway Arima Line to terminus Arima Onsen. (Note that express services terminate at Arimaguchi, so you'll need to switch to a local train.) Alternatively, if you're coming in on JR, get off at Kobe's Sannomiya station, take the Kobe Subway to Tanigami (谷上), and connect there to the Kobe Arima Line. In either case, make sure to transfer at the Arimaguchi station onto the (one stop only) Arima Onsen line. The whole trip will cost around ¥900-1000 and take about half an hour.
The fun way is to take the Hankyu line to Rokko station and continue up Mt. Rokko on the cablecar (10 min, ¥570/1000 one-way/return). Connect with the bus loop line (10 min, ¥250) to the Rokko Arima Ropeway, then take the ropeway down to Arima Onsen (12 min, ¥980/1770 one-way/return). The Rokko Arima Katamichi Jōshaken (六甲有馬片道乗車券) combination ticket (¥1700 one way) works out marginally cheaper, and also lets you ride around the top of Mt. Rokko. This route is more scenic, but best avoided with heavy luggage.
It's quite easy to get around town on foot, though the uneven terrain may prove tiring to some. The JR train station has a large map of the area, but please note that it is neither to scale nor oriented to the north.
Arima Onsen is famous for geisha performances, held at the many yearly festivals. Note that in the local dialect, geisha are known as geigi (芸妓).
Arima Onsen being one of Japan's famous hot spring towns, it goes without saying that one should pay a visit to one of the many baths. There are three public baths (admission is between 500-700 yen) and a number of private ones. Be aware that Japanese onsens are sex segregated and bathing is done in the nude. Make sure you brush up on your onsen etiquette before you go!
Souvenir shops sell a variety of items, including bottled hot spring water and a plethora of spa and skin care goods using minerals from the hot springs. Arima Onsen is also famous for it's woven bamboo baskets and writing brushes.