Edinburgh is vertical. You will underestimate distances that look short on a map because they include a crag. Plan one hill thing per day: castle or Arthur's Seat, not both before lunch.
The Royal Mile exists to move people between the castle and the parliament and to sell them wool. Your trip gets better the moment you treat it as a corridor. Leith, the New Town terraces, and a proper tasting room are the city.
Where to spend your money
Book: castle timed entry, one underground tour, Britannia or a whisky tasting.
Skip: haunted pub crawls, hogmanay-priced mediocrity outside New Year, and any tour that promises the Highlands and Loch Ness before dinner.
The part nobody tells you
The weather can do four seasons before noon. Waterproof layer in your bag is not a stereotype, and Arthur's Seat in a whiteout is just exercise. Wait for a clear band of sky — it usually comes.