*CaltonAnd Carlton Hill for some iconic views.
Can recommend the Castle, underground tunnels (organised walking tour or ‘ghost’ tour - well worth it), boat trip under the Forth Bridge and out to Inchcolme Island, a stroll around the Meadows and nearby and generally just mooching around. Absolutely dozens of pubs and bars to explore, too many to recommend ! Arthur’s Seat isn’t far to go and not much of a climb but well worth it for the views. Best meal I’ve had in Edinburgh was at The Kitchin in Leith (as others have said, Britannia also worth a visit if out there). The Scottish Seabird Centre at North Berwick is a bit further out but great and easily reached by train (about half an hour from Waverley)