What connection string should I be using for my DNN (DotNetNuke) Installation?

Inside your web.config, please make sure your connection string looks similar to this:

    <add name="SiteSqlServer" connectionString="Data Source={local}; Initial Catalog=DB_NAME; User ID=DB_USERNAME; Password=DB_PASSWORD;" providerName="System.Data.SqlClient" />

or

    <add name="SiteSqlServer" connectionString="Data Source=localhost; Initial Catalog=DB_NAME; User ID=DB_USERNAME; Password=DB_PASSWORD;" providerName="System.Data.SqlClient" />

Of course DB_NAME, DB_USERNAME and DB_PASSWORD would have to match the username and database you've created through the control panel.