We recently came across a Vultr VPS that was not resolving names using Vultr's recursive DNS. This turned out to be a Vultr issue, which was resolved, but I wanted a way to add additional name servers in case this happens again.
Edit /etc/resolveconf/resolve.conf.d/base using nano/vi.
Add your nameservers like so. These will be use in addition to whatever the DHCP server provides.
nameserver 1.1.1.1
nameserver 8.8.8.8
Execute the following command, or simply reboot the system in question.
sudo resolvconf -u