You can certainly appeal against it. You must lodge your appeal with the Crown Court within 21 days of your conviction.
However, you haven't said how much the fine was and we don't know what your income is, so it's difficult to say whether you were fined correctly. However, as Bert says, six points is in accordance with the guidelines. What were you expecting?
If your appeal against sentence in the Crown Court fails the prosecution will ask for costs in the region of £700.