The MySQL site can be a little overwhelming, especially if you're a newbie. I'm not a huge fan of their navigation structure, although I am a big fan of their products.
I actually just downloaded MySQL 5.01, and did a video on how to set MySQL up, along with the driver and the GUI tools. I'll have it on youtube shortly. Anyways, I used the mysql-connector-java 5.1.6 JDBC driver.
On Sun's site, they have ODBC/J/Net/MXJ/Native & Visual Studio plugin drivers, which is indeed overwhelming. For your purposes, I'd say the J driver is the one you want - it's often referred to as connectorJ. 5.1 is solid, and specifically, I've used 5.1.6 without problems.