Lime Tree Avenue is the best place to look for Hawfinches. You might also be rewarded with close-up views of Nuthatches, and good numbers of Redwings on the lawn.
The bird feeding station at the side of the main path through the woods is a good place to see Marsh Tits, along with the more common woodland species.
Walk around the lake to see Great Crested Grebe, Goosander in winter, ducks, gulls and geese, plus Moorhen and Coot. Kingfisher is also a possibility, so keep a look out.