The best place to start is to buy a book. There are plenty of URi that I can post (see below) that will give you an introduction to both technologies, but I always think it's easier to read directly from a book at your own leisure.
Try these few URi to start you off...
http://www.devshed.com/Server_Side/PHP/DB_Basics/ - Database basics with Mysql and PHP.
http://hotwired.lycos.com/webmonkey/programming/php/tutorials/tutorial4.html - PHP/MySQL Tutorial.
You don't need any special software to run either technology as both are installed on Dreamhost servers.
MySQL databases are only available to higher-end customers, altough you may request a MySQL database as an 'add-on' to your existing plan.
I believe that Microsoft Excel and Microsoft Access are capable of exporting any database to CSV format (comma deliminated format). This then can be fed into a MySQL database, although results do vary.
A good program to help you manage your MySQL database using a web browser is MySQLman from Gossamer-Threads (http://www.gossamer-threads.com/scripts/mysqlman/). It's a free download, but a very very powerful program.
Hope this helps you to get started.