There are few ways to arrive Kinmen...
1. Fly from Taiwan...there are many domestic flights flying between Taiwan and Kinmen. Price are about NTD4,000. (round trip, USD 120+)
2. Take a ship...I think only those who live in Kinmen are allowed to take free government ship; otherwise, it takes about 20 hours to arrive from Kaohsiang.
3. From Xiamen, China: Please make sure you have valid visa for both sides.
The easiest way to move around in Kinmen is to take a bus, if you take from Shanwai, largest town in the east, to Jingcheng, in the west, it only cost you NT20; if you are a local, you don't need to pay anything for any public transportation. Kinmen is one of the richest county in Taiwan because of lucrative Kaoliang Wine. Now they introduce tourist buses, visitors can take it from certain bus station, there is a guide on bus for free tour. What a nice service!!