I'd only read/written to SD card files before. When you're writing to the SD card, you can use standard Java file I/O. But if you want to write to internal storage, you can't, because you're supposed to ignore where the files go.
You can do so with Context.openFileInput() or Context.openFileOutput(). When you do, the files will be in /data/data/your_project_package_structure/files/your_file. (yeah, that's two /data/s.) For example:
More details: official docs and someone's tutorial (which is a lot more useful).