The code below will copy the range selected in "Sheet1" (modify it to fit your sheet's name that holds the data to be copied), and then will paste it in Column A in the first available row in Sheet "All_Data".
Let me know if the code below (tested) works :
Sub CopySelection()
Dim Sht1 As Worksheet
Dim Sht2 As Worksheet
Dim xlSel As Range
Dim LastRow As Long
' modify "Sheet1" to your sheet source (where you make your Selection to copy)
Set Sht1 = ThisWorkbook.Sheets("Sheet1")
' sheet "All_Data" is your target sheet (Where you paste the selection)
Set Sht2 = ThisWorkbook.Sheets("All_Data")
Set xlSel = Selection
'option 1: find last row with data in Sheet "All_Data" at Column A
LastRow = Sht2.Cells(Sht2.Rows.Count, "A").End(xlUp).Row
'option 2: (less reliable) find last row with data in Sheet "All_Data" using UsedRange
'LastRow = Sht2.UsedRange.Rows(Sht2.UsedRange.Rows.Count).Row
' paste the copied range in Column A, the first row after row with data
xlSel.Copy Sht2.Range("A" & LastRow + 1)
End Sub