Where are my logs on my Linux VPS?

Typically on a linux VPS\VM that does not have Plesk most logs for services like http, ftp, and mail (depending on what kind of mail and FTP programs you install) can be found here:
# /var/log

If you have Plesk then in addition to the above you should also look in the following directory (specifically for maillogs):
# /usr/local/psa/var/log


If any logs are showing as constantly empty then you might not have the syslog service running or installed. 

To install it on a CentOS VPS first install yum (if you haven;t done so already, this step isn't necessary on a VM):
https://help.blacknight.com/entries/22936398-installing-yum-in-centos-vps

And then  run the following command:
# yum install sysklogd


If you have a Debian VPS\VM then you use apt-get instead of yum.  First you need to find the package name for the syslog service:
# apt-cache search syslog

Then install it using:
# apt-get install PACKAGENAME

where PACKAGENAME is the package name you get from the first command.