Arthur's Seat (or Arthur's Arse, as we re-christened it).
Calton Hill
You have to visit the castle, I suppose
Walk up or down the Royal Mile - tacky gift shop city
Outside bet: get the train to North Queensferry over the Forth Bridge and walk back over the road bridge.
RZ's a little unfair. Up/down the Royal Mile is a must - just keep your eyes above ground floor level:
Old and New Towns of Edinburgh - UNESCO World Heritage CentreThe castle is expensive to get into. Stop on the esplanade (the big open area in front of the castle) for the view. At the other end of the Royal Mile, a tour of the parliament is cheaper:
The Scottish Parliament: - Visiting Holyrood - Guided ToursThe
Museum of Scotland is fun, even if half of it's being rennovated. Free entry. Your challenge: find the secret snowman in the basement ...
Princes Street is best crossed, rather than walked along.
North again is/are
the Royal Botanic Gardens.
I like the
Dean Cemetry and
Dean Village. There's a path along the
Water of Leith to
Stockbridge (spot the
Shallow Grave exteriors).
The
National Galleries of Scotland have some nice things. There's a free bus between them.
Speaking of buses - a £2.50 "day saver" ticket on
Lothian Buses is a good-value way to get around.
If you have to get a train out, I'd recommend
North Berwick as opposed to Queensferry, unless windy suspension bridges full of traffic are your thing (OK, there's a wide footpath, but biking it's quicker and more fun).
There's a
farmers' market every Saturday morning. Fill up on
Stoats porridge.
RB's Embra-based these days, too.