Computers connected to router, but not to the internet?

I live with my two brothers and we each have a computer. We have a modem that is connected via cable and then a router that connects the modem to the three computers.

The speed is great and have no lag most of the time, but there is one problem. One very ANNOYING problem:

Most of the time when each of us turn on our computer, we are not connected to the internet, just the modem. It says "Limited Connectivity" and "Local Only." The only way to fix it is to reset the modem and router, and we're all set. BUT, sometimes we will randomly lose connectivity in the middle of the day and it will go to "Local Only" which doesn't allow us to open any web pages or play games that require the internet.

Now things get even weirder; it's random! At any given time my computer will lose connectivity, then maybe one of my brothers will, then maybe my other brother will. Our internet is NOT CONSISTENT!

We are connected to Google DNS, and have Xfinity High Speed Internet. Any help is appreciated!!!

Thanks.