Here is an example add-in button that does the basics of what you want:Imports ESRI.ArcGIS.Carto
Imports ESRI.ArcGIS.Geometry
Imports ESRI.ArcGIS.Geodatabase
Imports ESRI.ArcGIS.ArcMapUI
Imports ESRI.ArcGIS.ADF
Imports My
Public Class MyClass ' Your real class name will appear here
  Inherits ESRI.ArcGIS.Desktop.AddIns.Button
  Public Sub New()
  End Sub
  Protected Overrides Sub OnClick()
    Dim pMxDoc As IMxDocument
    pMxDoc = CType(ArcMap.Application.Document, IMxDocument)
    Dim pMap As IMap
    pMap = pMxDoc.FocusMap
    Dim pEnumLayer As IEnumLayer
    pEnumLayer = pMap.Layers
    pEnumLayer.Reset()
    Dim pLayer As ILayer
    pLayer = pEnumLayer.Next
    Dim pFLayer As IFeatureLayer
    Do While Not pLayer Is Nothing
      If pLayer.Name = "MY_LAYER_NAME" And TypeOf pLayer Is IFeatureLayer Then ' Substitute your real layer name
        pFLayer = TryCast(pLayer, IFeatureLayer)
        Exit Do
      End If
      pLayer = pEnumLayer.Next
    Loop
    
    If pFLayer is Nothing Then
      MsgBox("No Layer Found")
      Exit Sub
    End If
    Dim pFSel As IFeatureSelection
    pFSel = pFLayer
    Dim pSelSet As ESRI.ArcGIS.Geodatabase.ISelectionSet2
    pSelSet = pFSel.SelectionSet
    If pSelSet.Count < 1 Then
      MsgBox("No Features Selected")
      Return
    End If
    Using comReleaser As ComReleaser = New ComReleaser() ' Recommended for cleaning up cursors
       Dim pFCursor As ESRI.ArcGIS.Geodatabase.IFeatureCursor = Nothing
       pSelSet.Search(Nothing, False, pFCursor)
       comReleaser.ManageLifetime(pFCursor)
       Dim pFeature As IFeature
       pFeature = pFCursor.NextFeature ' Get First Feature
       If pFeature Is Nothing Then
         MsgBox("No Features Selected!") ' Just in case, but probably already taken care of earlier
         Return
       End If
       Dim lFIndex As Long
       lFIndex = pFCursor.FindField("MY_FIELD_NAME") ' Replace with your field name
       If lFIndex > -1 Then
         Dim fValue as String ' Substute the correct data type to match the field type
         Do While Not pFeature Is Nothing
            fValue = pFeature.Value(lFIndex)
            ' Do something with the value stored in fValue
            pFeature = pFCursor.NextFeature
         Loop
      End If
    End Using
  End Sub
  Protected Overrides Sub OnUpdate()
  End Sub
End Class