Gathering together news that matches your own interests is pretty easy if you use RSS feeds. If you’d rather have a more newspaper-like feel, NewsCred may be an appealing option.
NewsCred is designed to mimic conventional newspaper design, but drawing content from a range of sources on topics picked by you. You can search for topics that interest you, or add RSS feeds if you know there’s particular publications you like using. The coverage seems reasonably broad from an Australian perspective, and the interface is pretty straightforward (though we got it to hang a few times). You can even add your own editorials if you want to share the selection (and your views) with others.
One minor annoyance: links, when clicked, open in a separate tab, and there’s no way to switch that off in the settings. NewsCred is free to use (an optional paid service lets you use your own domain and add paying advertisements).