I would suggest you to start with a simple setup like you already told us: Eclipse + let’s say an easy to use tomcat.
If it comes to the server side you need to create a Servlet which allows to access other application data from your server in a format which you define or let’s say you + the clients define. I would suggest you to start with something you will be familiar like Json. For this I have a little “starting point” for you.
Clone that simple example for a server-side application:
https://github.com/DominikAngerer/java-GsonJerseyProvider
It already provides you a simple structure for Json which is used by a Framework like Ember and also you can easy modify the structure which is defined there.
Is the common practice to develop your code in Eclipse and then “ship” it to your server?
Yeah it’s totally common to do that. Write your Server Side code and export it as .war – tomcat or jetty for example can simply deploy them and you will be able to run them on these application server.
Do you know any good, non-specific tutorials for server-side Java? I’ve seen some tutorials for Google’s Cloud Platform and Netscape, but I don’t want to be dependent on their infrastructure.
I really love to work with Jersey – It’s really a to understand the code you will write. Maybe try the example provided by IBM developersworks: http://www.ibm.com/developerworks/library/wa-aj-tomcat/
For the beginning it may be a little of a overhead but after you tried the example (it should be running right away) I think you will get the idea of how to create a simple backend system with java.