One of the best preserved old cities in Europe, Edinburgh itself is a museum to its long history and Scottish charm. There's not many modern looking buildings, but that doesn't stop it from being a very modern city set in the dressing of centuries-old times. It's also very walkable and easy to get around, which is great if you don't have access to a car.
From the ancient castle to Holyrood, is the Royal Mile where you can walk along the same cobblestones the ill-fated Queen Mary had once trotted along. There are many fine pubs and bars littering around such as Bar Kohl, which boasts an incredibly impressive vodka list. You can pay a visit to Elephant House for not only delicious coffee and a fine view of the Castle, but to pay homage to Harry Potter. It's said JK Rowling wrote parts of the books there and the bathroom has become a bit of a love letter to the series.
If you're adventurous, you can test your mettle climbing Arthur's Seat to get an amazing view of the city and surrounding area, or if you're less into hiking, you get quite the similar view from Edinburgh Castle.
I found it to be both an exciting city but not as exhaustive as London could feel. There was always something to do, but there was less rush, less crowding, and even during the busiest time that is Festival Fringe, I felt at ease strolling along the Meadows or through Grassmarket and didn't feel like it was packed.
It's a great place to spend a few days or make as base for visiting the Highlands and other parts of the Scottish countryside. It isn't the warmest destination climate-wise, but the locals can be very kind and friendly, and all it's historic beauty and charm make up for the colder weather.
Paranormal buffs will be pleased to learn it's considered one of the most haunted cities in Europe, and you can tour the city's underground and learn quite a bit about it's supernatural activity as well as take other various walking tours.
Or for the less terrifying, August is Festival Fringe and a wonderful time to catch a bunch of up and coming talent performing at the various venues across the city, though some will say this is Edinburgh at it's busiest which can be quite your mileage may vary, but I loved being there during Fringe and am quite biased.
It's a gorgeous place to visit in the UK, and if you want something a bit quieter than London and more compact, but with a lot going on, this can be a fantastic destination for you.