Database development/admin - What exactly should I be trying to learn?
For development you should learn:


SQL including optimizations
DDL (table creations etc.)
plpgsql, PL/SQL, T-SQL (native database language to write procedures and functions)
good to know C, java, scripting languages
jdbc how it works 
storage - how db store data and which solutions are faster then another
...


For administration
all above, scripting languages is a must
7. everything about operation system that you will work with
8. storage maintain
9. a lot of measure possibilities how server works (monitoring)
10. installations, upgrades - OS and db system
11. archive possibilities
12 high availability possiblities
13.....

and for both, a lot more things

About tools you have to know how to work with console very good in case that tools cannot connect at current time to db but you can connect to server itself.
Other tools are usually database system specific.