If your database requirements are fairly modest, then SQL Server Compact is fine.
If you've created an .sdf file, then I assume you must have used Sql Server Compact anyway as SQL Server Express creates .mdf files.
However, it's just dawned on me that you should be using the classes in the System.Data.SqlServerCe namespace rather than the System.Data.SqlClient namespace which may be the reason for the latest problem.
So, I'd try:
string connString = "Data Source=|DataDirectory|\db01.sdf";
string query = "SELECT * FROM table01";
SqlCeDataAdapter dA = new SqlCeDataAdapter(query, connString);
SqlCeCommandBuilder cBuilder = new SqlCeCommandBuilder(dA);
DataTable dT = new DataTable();
BindingSource bS = new BindingSource();
bS.DataSource = dT;
dgv01.DataSource = bS;
You'll also need to add a reference to System.Data.SqlServerCe.dll if you haven't added one already.