Outside DNS/Firewall issues

I am not a networking professional. But I am assisting our Network admin. What are the best ways to troubleshoot public DNS and ISP routing issues. We are a small school district and during our break our outside DNS crashed causing issues with some of our online portals. Our firewall (sonic) support suggested a few changes to our outside DNS server and things started working on the portals. But now our access to most of our educational secure sites are timing out and/or getting DNS errors. What are best practices on how to troubleshoot these type of issues? Thanks...