DBA Job responsibilities vary by organizations. Anyways, You can check the following Document which lists the daily, weekly and montly tasks that are performed by DBA's in general. This Document is old but still holds good.
Those lists are good places to start. As was stated above, generic duties vary from site to site. The best advice is to be proactive on your systems and dig into them to insure you know how they are set up and running so that when a problem comes up you don't have to spend time looking to see how it was.
And for your backups I cannot stress heavily enough to verify the backups were valid. I knew a site where all they did to verify the backup was to look at the byte count for the tape backup. Then when they had to restore from tape they then discovered that their backups were not any good and they could not restore from tape. From time to time take a backup tape and restore it to a test server to insure that your process is working correctly.
For the most part we sit around drinking coffee and verbally humiliate developers. Of course there is the occasional discussion on sports, and the ever popular Biggest Booger Contest of which yours truly is the reigning champion 2 years in a row.
Now where is my double mocha frappachino late’.