Daily activities of the DBA depends on which kind of organization you work: it can be very different from one company from the other one.
I would suggest you to read first:
1. Oracle® Database 2 Day DBA
2. Oracle® Database Administrator's Guide
A good starting point can be to learn how the DBA team does backups and restores: this is one of the most important thing that a DBA must really know to do in theory and in practive.
As a DBA I usually take showers, eat, drink, speak with people... :-)
I think it all depends on your company, environment, team, etc. I mean, you'll not do the same things whether you're the only DBA or if you're part of a team, or head of a team.
What I would do on the first day of the job:
. Say hi to the team, and get to know as many people as possible. As a DBA, you're not likely to stay everybody's "best friend" for long ^^
. Set up a little "welcome aperitif"
. Stick the Network admin in order to get an overall idea of the network
. Check with the Sysadmin(s) of the box where the databases(s) run how they configured the servers, how they manage it (backup, patch, ...), and if they're using Windows, convince them to migrate to Unix :-)
. If there's a DBA team, check with the team manager where are the databases, versions, ... if there's no team, just use your boss and colleagues provided info to find where are the database(s).
. Connect to one and every database and gather info : Log mode, jobs, size, statistics, ...
Well, these are just some possibilities. I certainly have forgotten lots.