Sysadmin's understanding isn't about memorizing apt-get or yum install, is mostly about ./configure --help ,)
After you will be able to build from the sources full LAMP stack, understand dependencies and cross-references, realizing the amount of crap^Wcode apache+php uses - this will be something to start with (integrating Informix or Oracle support is a bonus,)
The next step is to understand and internalize what is the difference between mod_php and fastcgi configurations - why they are different, how to enable caching, how it works, what are persistent connections to DB, etc.
The next level - be able to visualize in your mind the flows of data, and follow requests through all the layers - OS, http-server, fast-cgi transport, scripting engine, database-connection, table storage (realization that MyISAM engine or MongoDB are using table-level locking is a sign that you're on the right way).
Then, you must be able to separate the flows, partition the data, at least understand that /var/log and /var/db/mysql on the same disk slice is a sign of stupidity.
Then comes access patterns - how often the data are accessed, when changes occur, how often they are synced, etc (here comes caching and replications, then sharding).
This ability, to understand, trace and see patterns makes one sysadmin.
After that you might say that you kind of sysadmin.) Learning ssh is nothing. Googling is enough.