Some cases:
(a) static network settings in /etc/network/interfaces; then /etc/resolv.conf sets dns. Just edit to your needs.
Dynamic:
(b) plain dhcp: you can superseed (overwrite) the routers dns "suggestion".
(c) "network managers" there is a zoo of different tools. e.g. NetworkManager, connman, resolvconf, and what not; all with different gui and console tools for making changes.
Addional hardware:
(d) have a personal router - not the one from the Internet Service Provider. Set it up to your needs and any client will have the wanted dns server automatically.

a,b is good for static LAN, c for wireless-LAN or more dynamic setups (and for static LAN too, if you like).

Edit: Some typo. And the list is not complete, of course.

I am trying to change the DNS server permanently.  How do I go about this?  I have little to no idea on how to do this.  I searched on internet how to do it on Linux, but they all seemed to be only for the distros that are based on systemd.  Any help would be appreciated. 

Thank you.