Microsoft Edge also has a built in reader, like Firefox

I use the built in reader in Firefox.

Reddit would be a good example, lol.

Use landmarks. Screen readers can skip from landmark to landmark (banner, search, article, etc). Here's an example of an article on one of the easier to use news websites I use:
http://tvo.org/article/current-affairs/the-next-ontario/why-ontario-must-take-action-on-the-energy-east-pipeline

I work with ARIA pretty frequently and from what I've learned from speaking with people is to use the ARIA-Live function lightly.

It can be really distracting to have a users reading process interrupted by non-important information.

I usually test on Apple's iPhone for screen-reading, but as others have mentioned, Firefox and Microsoft Edge are also good