Knowing who to talk to about a given issue (for a large company). Since you say "new career", I assume you're starting a new job somewhere. I work for a fairly large organization that's heavily silo'd with regards to responsibilities. When something goes wrong, I often have to work with 4 or 5 departments, each of which has some folks who are helpful and folks who are 'blockers'. Figuring out who the right folks to talk to for a given issue can make the difference between solving something in minutes or hours (days). Likewise, making sure folks know my responsibilities and when to bug (or, more importantly, not bug) me.
Know your domain tools and always be looking for new (to you) ones. If you're new and in the linux sysadmin domain, you'll be doing a lot of "oh wow, I didn't know about XYZ". Find the old stuff that's been around forever (and likely installed everywhere) before jumping on the new hotness that'll require big changes to your systems to install. I like HN for some of this. Read the comments for the 'vim/tmux/bash/cron/git guide' entries that get reposted every few weeks. I guarantee somewhere in the comments, you'll find something you didn't know that'll help.
Scripting tedious tasks is huge. Learning the bits of shell/perl/python/ruby/whatever that interact with the tasks you do day-to-day can be very beneficial. Automate all the things.