The City Sightseeing Tour will help you to get your bearings, you can hop on and off as you like.
Edinburgh Castle, Camera Obscura, Scotch Whisky Heritage Centre and St Giles Cathedral are all in the same vicinity.
You could then take a walk down the Royal Mile to Holyrood House - you get great haggis,neeps and tatties in The Wee Windaes on The Mile!!
There are a couple of galleries in Princess Park, good place for a packed lunch if it's dry.
Scottish Parliament is supposed to be quite good, if you fancy a walk there are great views from Arthur's Seat.
Personally i don't think much of the shopping in Edinburgh, but i do live in Manchester and we're a bit spoiled here. Although i enjoy the walk around the Grassmarket from the Royal Mile - a number of quaint shops and there is a fantastic cheese shop on the way round.
I've been to Edinburgh 7-8 times over the past few years and have yet to make it out to The Royal Yacht Britannia so i don't know if it's worth the trip - but i believe you can get a bus.
The Zoo is out of the city and does take a while to get around, it's good but you might find it hard to get much more in - you could save it for the second day if you have time. If you have some time but not a lot (on your second day) The Royal Botanic Gardens are lovely, esp if it's a nice day - it's not free.
You will be spoiled for choice when it comes to food, but if you like Italian Deli's and are anywhere near Valvona and Crolla (they have a website but i think it's on Elm Row) - it's like an emporium and they have a lovely eatery round the back.
There is also a fantastic Italian takeaway in Stockbridge called L'alba D'oro, and next door is a great fish and chip shop. It's run by the same family.
Enjoy your time there and don't forget your camera and your comfy shoes!!