We have an open jaw though Heathrow in late April. We arrive on a Monday afternoon and need to be back for an 8:30 am flight on a Saturday. We were thinking of flying to Edinburgh and then doing a few day trips. We want to see the city and also get in some good scenic hikes. I want to focus on a couple things since we don't really have time for much more. Could us some advice on a day trip or two and some "can't misses" in the city.
Edinburgh castle is the most popular attraction in Edinburgh, but without more info, it’s very hard to advise you. Perhaps you hate castles, I don’t know.
Roslyn chapel is a short day trip from Edinburgh. There are no can’t miss attractions. It’s all about what you are interested in.
Hikes in the city of Edinburgh would be Arthur’s seat or Carlton hill. Both give great views of the city.
We enjoyed the Royal Yacht Britannia. It is a short bus ride to get there. Here is their website: www.royalyachtbritannia.co.uk
Edinburgh Castle is worthwhile, too.
As diveloonie said, Arthur's Seat and Calton Hill are nice walks in the city. Calton Hill is a pretty short walk. Arthur's Seat will give you nice views of Edinburgh.
There are many, many "can't misses" in Edinburgh. Best thing to do is get hold of a good guidebook, like Lonely Planet or Rough Guide, then decide from there. Also keep track of events happening in Edinburgh around the time you're there. There may be concerts or parades that you'll want to attend.
As you only have four full days, you want to pick and choose carefully. You could spend half a day alone in the National Museum of Scotland. The art galleries are wonderful, and the walk from the castle down to Holyrood Palace, followed by a hike up Arthur's Seat is good for another half day.
For scenic hikes, in addition to those already mentioned (Calton Hill and Arthur's Seat), there is a good hike in Roslin Glen, if you're planning to visit Rosslyn Chapel, and several excellent hikes in the Pentland Hills, just south of Edinburgh. Also the Water of Leith Walk, from Balerno to Leith (although you can join it anywhere along the way). I'm assuming that you won't have a car; all of these hikes/walks are accessible by bus.
As far as day trips, Glasgow would certainly be a good one, or Stirling, or Dundee. Falkirk as well, for the Falkirk Wheel, the Kelpies, and Callander House. There's a good hike along the Forth and Clyde Canal from the Wheel to the Kelpies, or you can catch the number 3 bus. All of these locations are easily reached by train or bus from Edinburgh.
Most of the sights in Glasgow are within walking distance of Queen Street Station: Glasgow Cathedral, the Necropolis, People's Palace, Tenement Museum, Glasgow Green, etc. The Kelvingrove Museum and Art Gallery can be reached by bus; it's near the University.
In Stirling, you can visit the castle and the historic area around it. Central Dundee, near the train station, has the Discovery Museum, featuring the RRS Discovery, which Scott sailed to the Antarctic, and the newly opened Victoria and Albert Museum.
If you're at all interested in golf, St. Andrews can be reached by train from Edinburgh to Leuchars, then a bus in to town. Even if you're not interested in golf, the town itself is a great place to walk about, and you can walk on the beach and explore the cathedral.
Hope that gives you a head start.
p.s.: I'm a bit confused. Is your 8:30 a.m. flight on Saturday out of EDI or LHR?
Do you have Rick Steves Scotland or Rick Steves Great Britain? They each have the same very good chapter detailing Edinburgh attractions, including some scenic hikes within the city itself. Edinburgh has an amazing physical setting, so walking and hiking there is very picturesque.
Edinburgh itself can keep you busy for 3-4 days, easily. I was just there for 3 nights and didn't get to everything on my wish list, and I didn't even want to see Edinburgh Castle! There are also lots of day trip possibilities, both on your own and with organized tour companies such as Rabbies.