There is not much difference at all in using a Mac than other platforms for working with JDBC.
JDK
You need an implementation of Java, a "JDK". See my answer to another question about installing Java 7 and Java 8 on a Mac, including links for downloading a JDK. Before doing the database stuff, be sure this works in the Terminal.app program: java -version
JDBC Driver
You need a JDBC driver specific to your particular database engine.
H2 Database
I suggest trying the H2 database rather than MySQL, only because it may be a gentler easier way to get started. H2 is pure Java, rather simple in terms of installing and administrating, and free-of-cost. H2 comes with its own JDBC driver. The H2 web site has a quick-start page and a tutorial page. It is not written explicitly for Mac OS X, but you should be able to "translate" as needed.
Oracle Tutorial
Then follow the JDBC tutorial provided by Oracle.
Also: StackOverflow is for specific questions on programming, not general or wide-ranging discussion.