Since google reader is leaving us, I was looking for a replacement for some time. Several notable alternatives are: feedly, the old reader, newsblur. I tried all of them but unfortunately none of them keep up to my expectitation at least at their current version. Feedly has nice interface, but I don't like their mobile app interface. The old reader is loading very slow. Newsblur free version limit the number of sites you import.
Setting up a server for personal use is not something internet guru can do. With lots of PaaS and IaaS options available, you can easily set up a server with enough functionality for simple usage like -- a RSS reader.
Tiny tiny RSS is a very good server-side AJAX-powered application that runs on you own server and can be accessed almost anywhere (browser, Android). It's also free to use and comes with plenty of functions. Fever is another application hosted on your own server. It has a "what's hot" feature, which meatures which topics in your feeds are discussed most online. This gives you a sense of what is important and what is not.
I installed Tiny tiny rss on my Amazon EC2 server two months ago, and I really like it.
After some customization borrowed from this post, it looks very much like google reader.
You can also install tiny tiny rss on heroku. Here is a post describing how to do that, and here is a script made by the same author. I've never tried this method, but since we are only using one dyno with free tier postgresql, so it's basically free hosting. That's also another one nice advantage of self-hosting application -- you don't need to worry about they shut down on you or bump up the price.