First off how did you know that you needed a new router? There could be a number of thing to try before getting a new router, so I hope you didn't just run out and by one. I would recommend plugging all of the cables into the setup like it was. Then unplugging the modem, and router. Next wait about 5 minutes, then plug the modem back in, wait for it to power all the way up, then plug the router in. Next see if you can get online with any of the computers.
Second: If you still can't get online after restarting everything. Then I would recommend bypassing the switch, take it right out of the equation. Unplug all the computers from it. Then Try plugging the computer one by one into the ethernet cable that runs from the router. See how many you can get to connect the the net. If you get any of them to connect, the we can troubleshoot some other issues. If not there is another thing to try. This will let you know if you have a bad switch.
So far you have tried to do a power cycle, and remove the switch. Now I want you to hit the restart button on the router, it will either be on the back or the bottom. You may need a pen or paper clip to reach it. Once you find it hold it down for 10-30 seconds depending on what your instructions say. Then let the router do its thing. The lights will probably shut off and then flash and turn back on or something. Once the router is all set try to connect one of the computer and see if you can get online.
Try all of those ideas, in that order and hopefully you will be back in bussiness. If not then post back what you find out.
Cheers!