While flying is a good way to get from one end of Japan to another, it's sometimes made less convenient by the time and cost of traveling between the airports and the cities they serve. In Kobe's case, this just isn't an issue: and not only is it easy to get to, it's a hub for some of Japan's budget airlines, with reasonably priced flights available to cities across the country.
Located on Port Island, it takes less than twenty minutes on the PortLiner monorail from Sannomiya station in the center of Kobe, with easy connections across Kansai or to Shin-Kobe bullet train station. There's also a regular ferry to Kansai International Airport which takes just half an hour, and hourly buses to Tokushima in Shikoku, with a few stops along the way.
The terminal building is nice and compact, with the check-in counters located in a spacious central concourse. On the upper floor there are some souvenir shops selling a decent range of local goods and souvenirs such as Kobe cakes, Kobe curry, chinese dumplings, and ice cream. There are also a handful of restaurants and cafes if you want a meal or snack before your flight, or a small rest area with tables if you've brought your own food, while once you go through the security gates there's a PC area, a lounge and more snacks. Going upstairs also takes you to the broad observation deck, which is close enough to the gates and runway to afford a good view of flights taking off and landing.
There are a few flights operated by ANA, but the airport is used primarily as a hub for Skymark Airlines, Japan's main budget airline (or Low-Cost Carrier, or LCC), with cheap flights all across the country, from Hokkaido in the north to Okinawa in the south. There are also a handful of flights operated by Solaseed Air, another budget airline connecting cities in Kyushu with Kobe and Tokyo which sometimes offers discounted fares for tourists from overseas, and by Air Do, a budget airline focused on Hokkaido.
Close to the city, well connected, and easy to negotiate: if you're traveling to or from Kansai, Kobe option is an option definitely worth considering.