What are the concepts that need to be gone through for MySQL DBA?
Storage Engine
You should be very fluent with the use of two main storage engines: MyISAM and InnoDB
You must learn :

how data and index pages are laid out
how table and indexes are cached
how table formats can make significant differences in reads and writes
how table-level and row-level locking are handled
how queries that mix both storages engines affect performance
how to backup data from each storages engine

MySQL Replication
You should understand such things as

What are the I/O and SQL Threads
When log-slave-updates is necessary for a slave
What allows you to write on a read-only slave
When it's appropriate to use MyISAM/blackhole on a slave for tables that are innodb on the master
How relay logs are populated
How to recognize slave errors
How to filter out/fileter in SQL for specific databases and tables
Why run backups for slaves rather than masters

Troubleshooting and Tuning Queries

Learn how to run EXPLAIN on queries
Learn how to create indexes to properly speed up queries
Learn how to tune queries
Learn how to activate the slow log
Learn how to convert the slow log into MyISAM

INFORMATION_SCHEMA
You should learn how to query the INFORMATION_SCHEMA database for table info, row counts, last access info, and things like these.
Third Party Tools
The best tools that you should embrace are the following:

MAATKIT
Percona Toolkit (fork of MAATKIT also from Percona)
XtraBackup
MONyog / SQLYog

This is not a comprehensive list. However, it should be enough for anyone to be proficient as a Developer/DBA.
Here is Oracle's Certification Exam Descriptions that provide lists of skills within MySQL to have and know: EXAM1 and EXAM2
IMHO Here is a good set of MySQL Books to learn from