you mentioned core files, there are utilities to analyze them when a program dumps core, but from what ive read most people wont find too much use with the actual core file, unless you are a programmer. now, finding out why a programed dumped core is more useful, check the logs of that program and so on. the directory /var/log/ is probably where most of your log files will be kept. you will find your self in that directory everyday, look for anything out of the ordinary...and also use the last command daily just to make sure there are no 'strange' logins.
to yls177, daily things an SA does would be to go through the log files, and just generally making sure everything is working as it should. IMO the great thing about every unix is that once youve configured your box, thats it, if its doing its job now, theres not too much reason for it to stop doing its job later----given that you monitor log files well to watch for any problems, intruders, etc...another thing is to make sure you monitor available space on your system, getting rid of un needed programs and making sure any services that are not in use are turned off.