Here is a general idea of what I believe might be useful.
(I'd love to hear from some other experienced DBAs and consultants as well - Mark Powell, Billy Verreynne, Don B., Nicolas, Werner, to name a few)
Step 1: Take a backup. And verify that it is a good one. And get an interim backup/.recovery strategy in place.
If the previous DBA left, was hit by a bus, or was shot ... your first responsibility is to ensure recoverability of the database.
Step 2: Find out everything you can at a high level and document it.
Start with the basics:
- any userids and passwords (OS, user and DB) that you can determine;
- any files involved (tablespace datafiles, control files, log & archive log);
- any setup or configuration information (eg: nonstandard parms from alert.log);
- software & patch level involved (OS, Oracle, vendor/developer);
- user community - who, who is key, how to get on their good side;.
- architecture ... how does data get in, how does it get reported;
- know the politics!
Step 3: What are the objectives of the database and it's application(s)?
- backup, recovery, disaster recovery requirements;
- current backup techniques;
- what is the lng term future of the application?
- who holds the funds for application and DB maintenance?
- know the politics - better!
Step 4 - develop a plan for ongoing maintenance based on the above.