Tiny Tiny RSS is a open-source RSS reader that you can install on any LAMP server (Linux/Apache/MySQL/PHP). I installed this yesterday and it wasn't too complicated. Lifehacker has a tutorial, but one doesn't need any Linux or command line experience to install this application. You do need to be comfortable with cPanel and FTP software.
Download the tar to your hard drive and extract.
Open the Web Host Manager for your server, and go to the cPanel for the domain onto which you are installing the application. Then go to the MySQL Database Wizard. You will need to create a new database, a new user for that database, and create a password. Then give the user all privileges. Make sure you note these all down, and then you can sign out of your Web Host Manager.
Next, open FTP software (like Filezilla). Rename the folder that you extracted from the tar to tt-rss, and upload the folder (and all of its contents) to the public_html directory of your domain.
After upload is complete, visit the install page at http://YOUR-DOMAIN.com/tt-rss/install/. Enter the database settings you created in the MySQL wizard, and hit the test button. If it was a success, initialize the database. The app will provide you the contents of a custom config.php file. Copy this text and paste it into a new file in a text editor. Search for SIMPLE_UPDATE_MODE in the file and change its value to true so that the feeds will update every 1/2 hour while you're using the app. Save the file as config.php and upload into the tt-rss directory.
Now you can go visit /tt-rss/ and log in initially with
password. You can go into preferences to change the admin password, create users, and so on.