This app was mentioned in 14 comments, with an average of 2.21 upvotes
Tiny Tiny RSS. You need a server to use it, but this way you can read the feeds with any browser or with the app.
Official one: https://play.google.com/store/apps/details?id=org.fox.ttrss News+ Plugin: https://play.google.com/store/apps/details?id=com.noinnion.android.newsplus.extension.ttrss
Both worth the money.
This one: https://play.google.com/store/apps/details?id=org.fox.ttrss&hl=en_GB and unlocker for it: https://play.google.com/store/apps/details?id=org.fox.ttrss.key&hl=en_GB But tbh, I never tested others, as there was only this one, when I started with tt-rss.
It never left for me. I figure if a news source doesn't have an RSS feed, it's not worth me following it.
I've found it's harder to find the feeds lately. If there is one (usually there is) it's not advertised, but if you check the source code you can probably find an XML or RSS link in there.
There are now options to limit your exposure to 'monetization' even more. I used to use Google Reader, then feedly. But feedly I'm sure tracks users' feeds/read, etc for their own advertising gains. Plus, I was looking for a good filter editor for feeds and feedly charges for that.
If you're tech savvy, you can download the open source Tiny Tiny RSS, run it on a vanilla PHP/MySQL server and start setting up your own RSS feed reader. There is even an android app for it. There is a small fee for the unlocker, but it's worth it as it's pulling data only from your own custom servers. It works great and includes RSS filter editors among other features.
Install Apache or nginx, PHP and PostgreSQL, then install Tiny Tiny RSS. Now you can subscribe to RSS feeds from your local browser or use the app or one of the alternatives. Also enable TLS and basic auth in your web server, get a domain name with dynamic DNS, get a TLS cert from letsencrypt and open a port in your router.
Isn't this one maintained by the project owner of TT-RSS?
Setup your own Tiny Tiny RSS instance, then install both the Tiny Tiny RSS trial app, and purchase the unlocker key by Andrew Dolgov.
All three software is Free Software, and you can and should setup HTTPS on your web server instance. In this way, you own the data, and no one between your mobile device and the HTTPS server can snoop on what you're reading.
I like Tiny Tiny RSS (1.271) https://play.google.com/store/apps/details?id=org.fox.ttrss
I have the server installed on a public server I own, I don't have to rely on any service going away
All the recommendations are great here (Palabre being my favorite) but if you have some basic webdev skills, you can roll your own using TinyTinyRSS. Going this route gives you privacy as it's all on your own server, and it allows filters which other services like Feedly you need to subscribe to. The filters was the main thing that made me go with this open source version, and I'm quite happy with it.
On desktop you'd view it using a web browser pointed to your own server.
On android, they have an app. Buy the one-time unlock code and you're set.
This setup requires a bit more effort, but if you're a webdev and/or have web server space, you can roll your own RSS reader. I went this route because I, too, wanted a way to filter our spam RSS posts while keeping the good stuff, and all the services (feedly, etc) charge for that feature.
It costs a little for the app and requires some minor tech knowledge to run the software on your own host, but this keeps everything on your own server, so no privacy issues with third party services like Feedly, etc. and it gives you the filtering feature which all the other services require you to pay for.
Good luck!