I believe there are two methods to accomplish this, would just involve adding:

Option 1: Grabbing the Next Control if an Enter KeyPress was performed

In the properties of your form, set the KeyPreview property of the form to true.

The code below will capture your "Enter-Press" event and perform the logic that you are looking for:

private void [YourFormName]_KeyDown(object sender, KeyEventArgs e)
{
    Control nextControl ;
    //Checks if the Enter Key was Pressed
    if (e.KeyCode == Keys.Enter) 
    {
        //If so, it gets the next control and applies the focus to it
        nextControl = GetNextControl(ActiveControl, !e.Shift);
        if (nextControl == null)
        {
            nextControl = GetNextControl(null, true);
        }
        nextControl.Focus();
        //Finally - it suppresses the Enter Key
        e.SuppressKeyPress = true;
    }
} 

This actually allows for the user to press "Shift+Enter" to go to the proceeding tab as well.

Option 2: Using the SendKeys method

private void [YourFormName]_KeyDown(object sender, KeyEventArgs e)
{
  if (e.KeyCode == Keys.Enter)
  {
     SendKeys.Send("{TAB}");
  }
}

I am not sure if this method is still commonly used or may be considered a "hack"? I would recommend the first one, but I believe both should work for your needs.