Initial account configuration

Hi,
I woud want to configure an account of a linux host from one of the following connection modes :
1 - public key authentication with become sudo mode
2 - if failed, public key authentication with become su mode
3 - if failed, password authentication with become sudo mode
4 - if failed, password authentication with become su mode

What is the best method to do this ?

And when I want to connect to a host with my public key when this one does not support this mode, the connection loops. How can i solve it ?

Thank