All the logs are under /var/log/, once you have spent some time examining them you'll realise when an entry looks a bit suss. Search google for some rootkit checkers among the more popular are chkrootkit and rkhunter. You may want to look at a log analyser to send a summary of whats happened by email every day (I use logwatch). Again google is your friend here.
the general: Thats a rather naive train of thought. Remember that there are several viruses 'for linux' they are mostly proof-of-concept, however there are a very small number knocking around in the wild. If you have a properly patched uptodate machine then these shouldn't affect your system.
Even then, there are still a number of reasons for having a virus scanner on a linux machine, take for example my setup. I have an email server on a linux machine that recieves all my mail. When each mail is downloaded, before it makes it to my inbox it is scanned by clamav and checked by spamassassin. So before any of my Windows workstations even access my mailbox emails with viruses have been deleted and spam has been filtered to another folder.
Further all user directories for the windows workstations are stored on the linux machine so late at night when server load is a minimum the virus scanner scans these home dirs.