Debian Linux Error Log Files too Big

I am not very much familiar with using an OS Linux server. I already experienced hardware failure which made me panic.
So I want to make sure all will be recorded before problems like that happen again.
I want to know how to check Linux Error Log Files under Cento or Debian type of server.
I already checked mine, and it was too big.
It is several megabytes bigger. Is there anything I can do to make it look smaller?

