Wester Craiglockart Hill is one of the places I'm planning to take my winter wildlife group in the next few weeks. I've taken birdwatching groups up this hill several times, and even if we don't see any birds, people are always impressed by the views across Edinburgh (in the photo below you can see Arthur's Seat and Salisbury Crags to the right of the photo).
This is a great place for nature of all types, making it ideal for a general nature walk. It's a Site of Special Scientific Interest for the unusual types of grassland found there. and the associated wildflowers (though most of these are not in bloom at this time of year!) There's also a good selection of fungi, including this one, which is (I think) some sort of funnel cap (if any one knows which species it is, please let me know in the comments section).