Should I split code into smaller functions? Advice on layout
I would basically write a method wherever you've written a comment on what a block of code does. This allows you to hide the specifics of a particular operation and makes it clearer which code paths are taken and what they do at a high level:

private Service1 GetService()
{
    return new TestProjectService.Service1();
}

private CommandMessages GetCommand(Service1 service)
{
    var command = new SimpleWebServiceApp.TestProjectService.CommandMessages();

    switch (WebServiceRadio.SelectedIndex)
    {
        case 0:
            command = service.GetData();
            break;
        case 1:
            command = service.GetDataLINQ();
            break;
    }

    return command;
}

private void DisplayResult(string text)
{
    LatestCommandLabel.Text = text;
}

private string GetUsername()
{
    return System.Security.Principal.WindowsIdentity.GetCurrent().Name;
}

private void SubmitData(string username, Command command)
{
    var dc = new TestProjectLinqSQLDataContext();

    var msg = new TestProjectCommandMessage
    {
        Command_Type = command.CommandType;
        Command = command.Command;
        DateTimeSent = command.DateTimeSent;
        DateTimeCreated = command.DateTimeCreated;
        Created_User = userName;
        Created_Dttm = DateTime.Now;
    };

    dc.TestProjectCommandMessages.InsertOnSubmit(msg);
    dc.SubmitChanges();
}

private void ReturnValue()
{
    var service = GetService(); // We don't care how we construct the service, just get it
    var command = GetCommand(service); // We don't care what how we get the command, just get it

    if (command != null)
    {
        // Display the result however it needs to be displayed
        DisplayResult(String.Format("Last Command Called (Using {0}) : {1}", WebServiceRadio.SelectedItem.ToString(), command.Command));
        string userName = GetUsername(); // We don't care how we get the username, just get it
        SubmitData(username, command); // We don't care how we submit the data, just submit it
    }
    else
    {
        DisplayResult("No Commands Available");
    }
}

protected void GetCommandButton_Click(object sender, EventArgs e)
{
    ReturnValue();
}