I have done research and can change the DNS server by editing
/etc/resolv.conf, but it seems like every time I reboot the setting is overwritten by something else...
How can I stop whatever that edit the file?
If your network address is being set by DHCP, for example your DSL router hands it the address, the it is probably getting the DNS servers to use via DHCP too. In this case, change the DNS settings on what-ever device is handing out the addresses via DHCP is the better way to go as it means other machines/devices that you plug in will get the right ones too.
You can change what the DHCP client requests by updating its configuration file. I don't know exactly where this will be on a slackware install - on Debian and similar it is
But if the name servers being given out by your router (or other DHCP server) are incorrect you should change the configuration there instead so you do not have to configure each client individually.
Slackware Linux (at least 13.0) uses the dhcpcd DHCP client to get dynamic IPs, as you can see by studying /etc/rc.d/rc.inet1. Quote from the manpage of dhcpcd:
However that option is not specified directly. Please see /etc/rc.d/rc.inet1.conf. Grep for the string DHCP_KEEPRESOLV. You must have that setting enabled for the config block corresponding to the correct network interface and its value must be "yes".