Select the first filtered cell then move onto the next filtered cell down

Try this code:

Sub SendEmail()
    Dim objOutlook As Object
    Dim objMail As Object
    'Dim RowsCount As Integer
    'Dim Index As Integer
    Dim Recipients As String
    Dim Category As String
    Dim CellReference As Integer
    Dim RowLimit As String
    'New variables.
    Dim firstRow As Long
    Dim lastRow As Long
    Dim cell As Excel.Range
    Dim row As Long



    Set objOutlook = CreateObject("Outlook.Application")
    Set objMail = objOutlook.CreateItem(0)


    Category = Range("D1")
    If Category = "Email Address1" Then
        CellReference = 4
    ElseIf Category = "Email Address2" Then
        CellReference = 5
    End If



    With ActiveSheet

        'Find the first and last index of the visible range.
        firstRow = .AutoFilter.Range.Offset(1).SpecialCells(xlCellTypeVisible).row
        lastRow = .Cells(.Rows.Count, 1).End(xlUp).row


        'Iterate through all the rows between [firstRow] and [lastRow] established before.
        'Some of those rows are hidden, but we will check it inside this loop.
        For row = firstRow To lastRow

            Set cell = .Cells(row, CellReference)

            'We are checking here if this row is hidden or visible.
            'Note that we cannot check the value of property Hidden of a single cell,
            'since it will generate Run-time error '1004' because a single cell cannot be
            'hidden/visible - only a whole row/column can be hidden/visible.
            'That is why we need to refer to its .EntireRow property first and after that we
            'can check its .Hidden property.
            If Not cell.EntireRow.Hidden Then

                'If the row where [cell] is placed is not hidden, we append the value of [cell]
                'to variable Recipients.
                Recipients = Recipients & cell.Value & ";"
            End If

        Next row

    End With


    With objMail
        .To = Recipients
        .Subject = "This is the subject"
        .Display
    End With

    Set objOutlook = Nothing
    Set objMail = Nothing

End Sub

I believe the Hidden property of a range is what you want. The following code worked for me:

Dim row As Range
For Each row In Range("MyTable").Rows
    If not row.EntireRow.Hidden Then
        ''' DO STUFF '''
    End If
Next

I have always found that using a For Each loop is a much cleaner way to iterate through data in an excel sheet. "MyTable" was the name I gave to the range of interest but if you prefer you can just enter a the limits of the range like Range("A1:D4"). Though I think it is a better practice to use named ranges as it makes your code more readable.

EDIT: To address your comment...

If you insert a row into the middle of a named range the limits of the range automatically expand. Though if your table is going to be the only data in the worksheet you can also use the UsedRange property of a worksheet object. For instance:

Dim row As Range
For Each row In Worksheets("MySheet").UsedRange.Rows
    If not row.EntireRow.Hidden Then
        ''' DO STUFF '''
    End If
Next

If all you have is the first row of the table you can expand this range to the full table using:

dim FirstRow as Range
dim LastRow as Range
dim myTable as Range
set FirstRow = Range("A1:B1")
set LastRow = FirstRow.End(xlDown)
set myTable = Range(FirstRow, LastRow)

And then use the same For Each loop as before. Hope this helps!