Hi Cathy,
Greeting from Peggy, Top China Travel.
As I've got you are puzzled about Gulangyu in Xiamen. Here I would like offer my answers one by one to you.
1) Is Gulangyu Island a convenient place to stay?
Yes, Gulangyu Island is a place with so many accommodation choices. And a recommended tour to Gulangyu will be 2 days 1 night there. It is convenient to get there from Xiamen downtown. You can choose BTS to the pier, and then take the ferry to the island.
2) How are the connections to the train and bus stations in Xiamen?
Gulangyu is connected Xiamen by ferry transferring. The pier is near a bus station. You can take bus to the pier and then buy a ferry ticket to get on the ferry. After about 5-10 min shipping, you will arrival Gulangyu.
3) Is it very touristy and busy?
It depends on the time you visited. If you visit there in high season such as National holiday, summer holiday, o any other public holiday, you will meet thousands of visitors and it will be crowded in everywhere. And in the rest time such as April, June, Sept, it will be better to visit.
For more information, you can check about Gulangyu introduction on /china-attractions/gulang-islet.htm
Peggy