The best way to travel in Kobe is on feet, especially in the centre among Kobe, Motomachi and Sannomiya . The shops in the street makes the distance even shorter.
If you don't want to walk and if you need to go a bit further there are 2 subways :The Kaigan Line that runs along the coast, and the Yamate-Seishin Line that runs toward the mountains. Then you have the train: The Hankyu, Hanshin and the JR lines cross Kobe in a west-east direction, The most busiest station of Kobe are the Sannomiya station , and the Motomachi station .
When you want to go more west or east, you can take the Bullet train (the Shinkansen) which you can take on the Shin-Kobe station in the north of the city. This train will take you to Tokyo in less then 3 hours time.
Also you can take a taxi, but remember taxi’s are expensive, and cabdrivers, can't always find the address good, so when you know where you need to be, write it down . Also there English is not that well.
Because Kobe is a Port, you can take a ferry to other places in the area, like Shikoku, Kyushu, and the Awaji-shima island. A city loop bus circles most of the city's sightseeing spots, stopping at both Shin-Kobe and Sannomiya stations.
For visiting the whole Kansai area, it would be handy to buy a JR-West Rail Pass. With this pass the pass holder has unlimited use of any standard trains in the Kansai area ( JR trains only). You can buy a card from 1 up to 4 days, and the price varries from 2000yen up to 6000 yen. For more information see the JR-West Website.