Monitor all newly spawned processes on a Linux machine

What do you want to know about those processes? If you can control who spawns the processes, strace -feprocess $SHELL will do.

If it's just an overview of their footprint, use process accounting (in the gnu acct package; use the lastcomm command), or higher-level tools like atop's logger mode. In the future, tools like trace and uprobes will be helpful to get detailed info out of the kernel.

auditd?

http://linux.die.net/man/8/auditctl

Snoopy might be the right tool for your use case.

If you need simple logger and you are NOT looking for security/auditing solution, then Snoopy might be it.

Disclosure: Snoopy maintainer here.