There's really no difference between using a mySQL database and a SQL Server or Oracle database for Windows Forms programming.
To connect to your mySQL database during development, you will need to download and install the MySQL .Net Connector on your development PC. Once you've referenced the MySql.Data DLL in your WinForms project, you can then use the MySql database classes to retrieve data:
string MyConString = "SERVER=localhost;" +
"DATABASE=mydatabase;" +
"UID=testuser;" +
"PASSWORD=testpassword;";
MySqlConnection connection = new MySqlConnection(MyConString);
MySqlCommand command = connection.CreateCommand();
MySqlDataReader Reader;
command.CommandText = "select * from databaseTable";
connection.Open();
Reader = command.ExecuteReader();
etc..
You will need to install mySQL server on a database server somewhere - one way to do this is indeed using wampServer (although it's a bit heavier than you really need - you definitely don't need Apache or PHP, although I guess phpMyAdmin is a good way to manage your mySQL database). Either way, you'll probably just want one central database on a single server, so you'd just install wamp there.