Root Shell, what Now

Hello how are you?

Well this time (my second post) I am posting with a real life situation, I am not sure if it fits here or in the OS category so please move it if it is necessary.

I have access to a linux lab that doesn't have a grub password, so you can bypass grub to get a root shell, and that brings us to my questions:

If i wanted to have access to a user that has root, or to give root privileges to another user without being noticed, what should i do?

And also, if I have root in a machine, and considering that they probably have the same password in the other ones, what else could I do?

I should mention that i do not have any intention of messing up their system and servers, I just want to know what else can i do with the information and privileges i have, and how deep i could get in the system.

Thanks

Really? You got root but you have no idea how to operate a *nix system? For shame, man.

What i do not know is what is the best way to keep that root, i know that creating a new super user wouldn't be to smart, so that is the first of my questions, what other options do i have to keep that root.

Edit and recompile a gnu program to include an exploit, or modify the kernel.