Best way to implement Client <-> Server <-> Database architecture in an Android application?

I tried directly connecting to the database using the JDBC driver for MySQL but my program is crashing so I'm not sure if Android "supports" the JDBC driver for MySQL.

Never never never use a database driver across an Internet connection, for any database, for any platform, for any client, anywhere. That goes double for mobile. Database drivers are designed for LAN operations and are not designed for flaky/intermittent connections or high latency.

Do I connect to a Java server program using sockets (or some other method of communication)?

It doesn't have to be Java. It just has to be something designed for use over the Internet. As Mr. King's comment suggests, Web services have been used for this for much of the past decade. For Android, REST Web services are probably the easiest to consume, since there is no built-in support for SOAP or XML-RPC. But whether the Web service is implemented in Java, or PHP, or Perl, or SNOBOL, is up to you.

Well, OK, perhaps SNOBOL won't be a viable option. :-)