I liked it! :>  It's no Microsoft SQLServer, but you get quite a bit for the
price (FREE! :>).  You will have a bit of a learning curve (maybe buy a
book, or get good with googling for help), but it's something you can
certainly do!  Maybe other's will suggest something different, but for my
first time, I started with the most recent stable version.  That let me
focus on the issues and think of them as my problem.  My main concern with
alpha/beta code or new languages/product is when I run into a problem, I
can't figure out if it's ME or the product!  So if you go with the stable
vers, chances are, not 100%, but chances are it's you.  That's a good thing,
you can learn more and more.  If it's product based, you're screwed until
the next ver, so you want those issues to be learning curve related.

Most recent stable version. :>

Good luck.

A few comments:

1. You should be either writing a GPL app or willing to pay a good bit
to use this database.  (This is legally and morally quite questionable,
as MySQL has happily accepted donated work under the premise that they
would stick with open-source licensing... but unless you intend to get
yourself sued, it's still best to stay away).  There are other
alternatives (PostgreSQL is a very good one) that don't carry this legal
baggage, and everything works exactly the same above the hood, except
that PostgreSQL supports a lot more powerful kinds of SQL that will
choke MySQL (at least unless you care to use an unstable alpha-release
version of their software).

2. If you are writing a GPL app, MySQL is a great option for a fast
database where data integrity doesn't matter (such as a chat board or
something, where no one considers their posts to be worth much if lost).  
However, it requires a lot of hoops, including non-standard syntax, to
get any kind of referential or transactional integrity.  Not recommended
for the non-expert.

You need two things to use most *any* database with Java:

1. The database server.
2. A JDBC driver.

The versions should match on the two.  You can download both from the
MySQL web site.  The JDBC driver needs to be accessible to the JRE,
which can be done in a few ways:  You can drop it in jre/lib/ext of the
system you plan to run on, or you can add it to your CLASSPATH
environment variable, or to the "-cp" option of your JRE, or if it's a
web-app, drop it in WEB-INF/lib relative to the web-app root, or for an
executable JAR list it in the Class-Path manifest attribute and then
drop the driver in the same directory as the executable JAR.

Then you'll need to install the database server, and then get started
with programming (there are plenty of JDBC tutorials on the web).