Connect a router to a routrt?

What does your router say? Go to system Preferences->Network tab and find the router IP address. Then put that IP in your router's password section and make sure all the router posts are set to turn all their ports on.

 

I think that your wireless router is configured as a primary router, and thus is causing the problems. It tries to handout IPs and configure other stuff that your primary router should do exclusively. This produces conflicts and crashes your routers.
In order to prevent this, just configure your wireless router as a bridge.
A bridged router doesn't interfere with the main routers configuration. It just provides the wireless network, but everything else is dealt with by your main router. It also makes network administration much simpler for you. You just have to configure (NAT, static DNS, etc.) on your primary router!

 

Well, that most certainly depends on your router model.
Just take a look at the documentation that came with it or google on how it's done.

If you're familiar with on how to connect to a router, you may as well browse through the settings and find the bridge mode somewhere.
Connecting to your router is done through an internet browser by entering the routers address into the address bar. Usually it's something like 192.168.1.1 for your main router. The router obviously has to be connected in order to open its preferences!

 

Have you restarted everything (this includes both routers and all computers/devices connected)?

 

This might work since the old computer might still be working with an IP adres given under the old setup.

Good to hear you figured things out with DNS etc. I have a second router as well. It took me a while to understand the instructions I found on Google. At the end it was easier than I thought. Just had to make sure the second router didn't give any IP addresses on it's own.