I read virtually all of them. Most of them provide RSS feeds, some are a bit hidden but it's Googlable.
I started with the basics BBC, NYT, Guardian, The Intercept for general news, The Conversation for science news without sensationalism, and some tech blogs.
Then I just read most of it, and follow some links to find new sources. Most of the times news start with "As reported by X", or just a link, so you can discover new sources like that.
You can also browse HN (and n-gate for the highlights) and Reddit to discover new sources.
If you add so much you can't keep up, remove some, or change the feeds into section feeds. Most online newspapers provide them. I miss Yahoo Pipes, and have yet to find a simple hosted alternative.
There is also RSS Bridge for sites without feeds (Twitter, Facebook), but I still haven't found the time to set it up.
You can also add paywalled sources to read the headlines only. You can mark them as read from the index.