For the Mikrotik's own DNS requests, go into the IP > DNS configuration, and specify whatever servers you wish to use. Some services such as dhcp client and pppoe client will automatically add DNS server addresses they may learn. You can disable this on both client types if you wish to just always use certain static servers.
The primary / backup behavior is automatic. If requests to one server do not receive any replies, or get rejected by one host, then the Mikrotik will try the other server.
As for hosts on the LAN, you can handle this in several ways:
1 - specify the dns servers in the DHCP scope
2 - specify the Mikrotik itself as the DNS server in the DHCP scope
3 - configure the Mikrotik to transparently redirect DNS requests to itself
Option 1 is the most normal - the servers specified in DHCP will be used by all DHCP clients.
Options 2 and 3 require that you enable the "Allow remote requests" option in the IP > DNS configuration.
NOTE:::
If you enable remote requests, MAKE 100% ABSOLUTELY CERTAIN that your firewall rules do not allow DNS requests to be accepted from the WAN interface, or your router WILL WITHOUT FAIL be discovered as an open DNS relay and then used as a reflector in DNS-AMP attacks. (a type of DDoS attack)
You can test whether your router is accepting remote DNS requests from the Internet by using a self-test website such as
http://openresolver.com/