The old-n-good way?

cd to the appropriate directory
vi the alertlog
Shift-g to go to the end -you want to check from the bottom up.
Check for Oracle errors from the bottom up -until you reach the point of your previous visit to this instance's alertlog - using ?ORA-
Investigate and document each error you find -take action when needed or document why you are doing nothing.

Once you get bored of doing this in a daily basis you will figure out how to put together a more automatic approach to alertlog monitoring.