FreshRSS is an open-source RSS reader you can self-host. I'm currently running it on my NAS, 'cause there's an easy package for it in the QNAP Club repo. The easiest way to set it up was with the SQL database, the other options took extra configuring from QApache.
FreshRSS gives you a web frontend you can keep in a tab in your browser. You can also enable support for the GReader and Fever APIs, which make it easy to use with other clients. I've found EasyRSS, which is on F-Droid, works well with GReader API enabled :)
Be warned that the documentation/help is very incomplete.
To automatically refresh the feeds you have to use a cron job.
On my NAS O use this line:
11 * * * * /share/CACHEDEV1_DATA/.qpkg/FreshRSS/freshrss/app/actualize_script.php > /share/CACHEDEV1_DATA/.qpkg/FreshRSS/FreshRSS.log 2>&1
Which updates the feeds at the eleventh minute of every hour and adds an entry to FRSS's log.