Have an amazing time! Edinburgh is one of my favorite cities—I was fortunate to go to grad school there and miss it every day. It’s easily walkable, though you’ll be trekking up and down a lot of hills. Hiking up Arthur’s Seat is a must-do, even though it’s a popular activity. But if you go earlier in the day it won’t be nearly as crowded, and the views are spectacular (particularly at sunrise and sunset). There are also other parts of Holyrood Park that aren’t nearly as busy, and wandering around there is so peaceful you’ll easily forget you’re in a city.
I’d definitely check out the botanic gardens, and while you’re in that part of the city stroll along the river to check out Dean Village (if you go on a Sunday, you’ll likely pass by the farmers market in the Stockbridge neighborhood). Or, head in the opposite area and you can explore the waterfront Leith area.
If you’re into Scottish folk music/history, then definitely check out Sandy Bell’s, which has live music pretty much every night. A lot of the people who play there are old friends who have been hanging out/playing with each other for decades. The Rooftop Terrace at the National Museum of Scotland and The Witchery are both great spots for (admittedly pricey) meals. The museum itself is also great for a rainy day, and the cafe inside is a great place to sit and read/get work done while also people watching. The Meadows is also a great spot for relaxing and people watching. While you’re in that area, you can also wander around the University of Edinburgh’s main campus area.
And if you’re looking to venture outside the city, Rosslyn Chapel isn’t a far drive away and it’s absolutely worth checking out!
Glasgow’s only a short train ride away, so that’s an easy day trip as well!
If you’re looking for more off-the-beaten-path places in London, I’d recommend checking out our guide to Hidden Trafalgar Square. It has a lot of fun, overlooked places to see!
The ruins of St. Dunstan-in-the-East are also a great spot to explore! They’re right near the Tower of London, but it’s so quiet and non-crowded that you’d never know a huge tourist attraction was nearby.