var
  f : TextFile;
begin
  //...
  try
    AssignFile(f, 'c:\FileName.txt');
    Try
      Rewrite(f);
      Writeln(f, 'First line.');

      Append(f);
      Writeln(f, 'I am appending some stuff to the end of the file.');
    finally
      CloseFile(f);
    end;
  except
    ShowMessage('Exception');
  end;
  //...
end;

Text := 
  'First line.' + crlf +
  'I am appending some stuff to the end of the file.' + crlf;
 
SaveStringToFile('c:\FileName.txt',Text);