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I happened to be in a situation where I am in charge for 100+ systems on top of my job as a programmer. Everything we run is Linux (Debian, Ubuntu, Centos) and gladly our sysadmin set everything with chef, so security patches can be applied quickly.
I would like to know how can I follow newly discovered vulnerabilities spending least amount of time.
As a programmer I am a fan of Gemnasium, which does this kind of job for ruby gems I use in my projects.
To be updated regarding your distribution your first stop should be the appropriate mailinglist and the advisory list from distribution
Debian
Mailinglist: https://www.debian.org/MailingLists/subscribe
Webpage: https://www.debian.org/security/
CentOS
Mailinglist: http://lists.centos.org/mailman/listinfo
Webpage: http://lists.centos.org/pipermail/centos-announce/
Ubuntu
Maillist: https://lists.ubuntu.com/mailman/listinfo/ubuntu-security-announce
Webpage: http://www.ubuntu.com/usn/
Further you can keep an eye on http://nvd.nist.gov/ or https://cve.mitre.org/
