For learning a new language, I've enjoyed code golf problems. You are unlikely to use any of the same syntax/approaches in "real-world" applications, but it gets you to use the compiler/interpreter a lot and you learn about all different parts of the language.

For a project though, I think anything that you know the domain of really well and are interested in so that you are happy to complete it/go back to it makes sense. I find I often like to build some sort of crawler (APIs or page data) that can then be consolidated and distilled into concise information.