NAT type strict

That should be all that is required. What I would first do is connect the Xbox direct to the modem and make sure your NAT is open/normal.

Next, when did you/they get the Xbox? Did it work with the old router?

Also, is it a Xbox 360 or Xbox One? 

Last, how is it connected to the internet?

I tried that and the NAT type was still strict, although I've heard that what the xbox displays isn't always updated.

We got it last Christmas and I believe it worked with our previous router which was an old Cisco router.

It's an Xbox One.

It's WiFi through the router, then modem, then Cox.

Wampa said:
I tried that and the NAT type was still strict

Then that is where I would start. Before you start changing settings on the router, you need to know if you can get open NAT direct to the modem otherwise you will be banging your head against a wall.

Also, where is it showing the NAT type? XBL test or in a game?

Well, that's where it gets odd. In the Xbox network settings it says Strict, but, for example, in COD, it says it's Open. They can play multiplayer and chat in games, but they can't set up parties with their friends.

We also get an error when we test the multiplayer in the network setting. It says it can't get a Teredo IP address.

It's been suggested that we factory reboot the Xbox, but the kids are deathly afraid they'll lose their progress on their games. I don't think any of that is stored on the Xbox, but they're not willing to take the risk.

BTW, thanks for taking the time to help me out! 

Try this? http://support.xbox.com/en-US/xbox-one/networking/multiplayer-connection-error-solution

Follow the troubleshooting through if the first one doesn't work. Need to make sure your router is getting a public IP address. Also, do you only have 1 xbox trying to go online at once?