Aye, there's the rub. Once you get things configured well, the life of a Backup Administrator can get quite boring!
As for the OP's request of daily/weekly tasks, that's going to depend on your site and what needs you have.
My site, for example has the following regular tasks:
Check Scratch pool media count
Check for failed jobs and investigate
Check to make sure Vault sessions run
Handle new clients / client changes
Less frequently:
Like JD said, patches patches patches.
Hardware maintenance (annual service/PM for library, etc)
Implement new procedures, features, etc
You will find that some tasks are universal (scratch counts, failure monitoring, etc) and some are site specific (vault sessions, etc). Really it depends a lot on what your company needs.
$0.02