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 (with Unix/Linux server) you may be able to use sudo command straight after login to change a user, before file transfer session starts.

FTP protocol does not allow this.

The SFTP and SCP protocols allow for this, but the actual method is platform dependent.

With SFTP protocol, you can use SFTP server option on SFTP page of Login dialog to execute SFTP binary under different user. With OpenSSH server, you can specify:

sudo /bin/sftp-server

Note that SFTP server binary may be located elsewhere (e.g. in /usr/lib/openssh/sftp-server).

With SCP protocol, you can specify following command as custom shell on the SCP/Shell page of Login dialog:

sudo -s

However typically you will not be able to provide a password for su (see remote command execution limitations). So you may be able to do the above 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=NOPASSWD: ALL

The above line is very benevolent, it allows user yourusername connected from anywhere (the first ALL) to run any command (the second ALL) as a root without being asked for password. So you should restrict it as much as possible.

For example with OpenSSH you may restrict it only to SFTP session by:

yourusername ALL=NOPASSWD: /bin/sftp-server

Note that as WinSCP cannot implement terminal emulation, you need to have sudoers option requiretty turned off (what is default).