How do I change user after login (e.g. su root)?

Easiest way is to allow direct login with the user account you need, if it is not allowed already. For accounts such as root, the direct login is typically disabled by default for security reasons. So when enabling it, have security in mind.

Particularly with SSH, you may want to keep password authentication (the most vulnerable one) disabled and use e.g. public key authentication instead. With OpenSSH server, you can do that by setting sshd_config keyword PermitRootLogin to without-password1.

In some cases you may be able to use sudo command straight after login to change a user, before file transfer session starts.

SFTP and FTP protocols do not allow this.

The SCP protocol allows for this, but the actual method is platform dependant (to switch to SCP protocol, see Session tab of Login dialog).

If you are connecting to a Unix/Linux server, you can specify following command as custom shell on the SCP/Shell tab of Login dialog:

sudo su -

You can do this only if you are allowed to do sudo su without being prompted with password. See sudo documentation to learn how to do that. For example you can add following line to sudoers file (/etc/sudoers):

yourusername ALL=(ALL) NOPASSWD: ALL

Note that as WinSCP does not allocate TTY, you need to have sudoers option requiretty turned off (what is default).