> also, would this be something to do with JavaEE instead of JavaSE?
Depends on what you need to learn and where your expertise lies, but yeah, nowadays the core business functionality *is* developed using JEE.
You can develop a core business functionality using web services i.e. expose your business logic using web services. You can then use a web-client developed using any one of the web development MVC frameworks out there along with a standalone client developed using Swing, SWT etc. In short, a web and standalone client for consuming a business functionality exposed as web services [which are nothing but just glorified servlets].
A serious web application might be a policy management system wherein you sell policies to customers, customers pay their premium, you generate an invoice etc. But then again, these *serious* business apps might just put you off and you'd lose motivation easily.
It'd be better to start off with something which you find fun. Something like an
online TODO list
application, a web application which helps you maintain a
list of all the cartoons you have seen till date, an online text to PDF converter [easy since you can use the libraries out there] etc. The possibilities are infinite...
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