以下是一个使用Visual Basic设计公用模板并连接数据库的示例代码：

Imports System.Data.SqlClient

Public Class DatabaseConnection
    Private connectionString As String = "你的数据库连接字符串"

    Public Function ConnectToDatabase() As SqlConnection
        Dim connection As New SqlConnection(connectionString)
        Try
            connection.Open()
            Return connection
        Catch ex As Exception
            MessageBox.Show("无法连接到数据库： " & ex.Message)
            Return Nothing
        End Try
    End Function

    Public Function ExecuteQuery(ByVal query As String) As SqlDataReader
        Dim connection As SqlConnection = ConnectToDatabase()
        If connection IsNot Nothing Then
            Dim command As New SqlCommand(query, connection)
            Return command.ExecuteReader()
        Else
            Return Nothing
        End If
    End Function

    Public Function ExecuteNonQuery(ByVal query As String) As Integer
        Dim connection As SqlConnection = ConnectToDatabase()
        If connection IsNot Nothing Then
            Dim command As New SqlCommand(query, connection)
            Return command.ExecuteNonQuery()
        Else
            Return -1
        End If
    End Function
End Class

请将你的数据库连接字符串替换为实际的数据库连接字符串。这个类包含了连接到数据库、执行查询和执行非查询操作的方法。你可以根据需要在你的项目中使用这个类来连接数据库并执行相应的操作。