East Lothian. I lived in Edinburgh for 12 years and didn't think of going East, but it's blimmin lovely. Do the castles – they're laid out perfectly for a car tour. From Edinburgh, go along the coast towards North Berwick. Stop in Dirleton, marvel, explore, get a weekend pass for Scottish National Heritage, lunch in the lovely pub across the road.
Then on through NB, and stop at Tantallon, marvel, explore, back in the car. Continue on to Tyninghame, stop in the tea shop for tea + scones. Then pop into Knowes Farm Shop for fantastic local produce – and feed the chickens.
Next castle is Hailes, Just the other side of East Linton, tiny and a jewel. Then take the path from Hailes south to the walk to the top of Traprain Law, Roman remains, astonishing views, marvel, explore.
Next stop I'd recommend is back into East Linton for beer + dinner in the Linton.
If you've got another day, drive up to Whiteadder Reservoir from Garvald and wander about there on the Herring Road tracks. Lovely, very very quiet, and back via Gifford for tea shops/pubs.
Also don't overlook the coast – the beaches at Belhaven (West of Dunbar) and Tyninghame (between Dunbar and North Berwick) are easy to get to, deserted once you get 15 minutes from the carpark, and just beautiful.
Here's the great little Walking in the Lammermuirs pamphlet from the Scottish Rights of Way Society.
http://www.scotways.com/scotways_assets/files/045_lammermuirs_map.pdf
And here's my pictures of the route (admittedly on a bike!) from a month ago:
http://www.flickr.com/photos/allankelly/sets/72157622343213717/
Cheers, al.