DNS Servers don't change very often. Nor should the gateway if your talking about the WAN side gateway. The ISP WAN IP address can and will change however the router will automatically pick that up. You don't have to do anything for the ISP/WAN side.
This is what you need to do:
After the modem is bridged and the router is online and everything is connected. Log into the routers web page. Go to the Status/Device Info/Wan Section. Make sure you are getting an ISP IP, Subnet and DNS addresses. Copy and paste the DNS only addresses you see here into the Setup/Internet/Manual section if these addresses are not automatically filled in. Sometimes they don't.
Ensure DNS relay is enabled under Setup/Network settings.
Disable Advanced DNS Services if you have this option.
Now for any and all devices you connect to the router, wired or wireless. Go into Setup/Network settings and make sure you are seeing the devices in the routers DHCP Client list. I recommend saving and reserving the IP address for each devices do that the IP address doesn't change and easier to manage and troubleshoot later on.
This takes care of the LAN side of the router.
Let the router restart an restart all of the devices. You should be good to go.