best walkable location.

Stay near the Royal Mile, halfway between the castle and Holyrood Palace. Its walkable to just about everything. Walkable from the train station. Here's the AirBnB we are staying at in August. When are you going? The good places are already booked in August because of the Fringe Festival

https://www.airbnb.com/rooms/3456257?guests=1&adults=1

Depends on what you are looking for!
When I visit big cities I prefer not to stay in the center. I like the peace and quiet of neighborhoods where locals live. I need my sleep!
We were just in Edinburgh for 7 nights around New Years and chose to stay in Leith, a suburb of Edinburgh. It's about a 15 - 20 minute bus ride into the center. Very cool neighborhood next to the bay with lots of independent pubs, coffee shops, repurposed buildings, and beautiful architecture. From Leith there are many bus lines that run into different parts of the city center. Most of them go every 15 minutes or so. I just used Google maps to figure out which one was best and hopped on the bus!
We stayed in an airbnb that is a loft in a converted warehouse. It was incredibly comfortable and a fun change from what we are used to. Here's the link if you want to take a look. Not sure how many people you will have with you but this place was great for our family of 3.

anywhere within a couple of miles of the city centre,Edinburgh has a superb bus service that is cheap and efficient so easy to get to any where you want to go.

if you make a short list of where you fancy staying and I will make comment on them as to area.

WOW thanks everyone! I am going in May Looking at the 7-10
Thanks Caroline

I agree with not staying right in the centre.
I lived there as a student many years ago, and it was busy even back then.
I've been back many times since.
Some areas I would consider ,( if not staying with family or friends), are Polwarth, Haymarket, Bruntsfield, Morningside, upper Leith Walk area, and the South Side such as Mayfield area.
In the area known as the "New Town", you could pick Stockbridge, Dundas Street area, Broughton Road area.
From Bruntsfield, Stockbridge and Haymarket you can easily walk into town in 30 minutes or so.
Edinburgh has a wonderful bus system, and during the day there is usually a bus every few minutes from a main road into "town".
A day pass costs 4 pounds, and you can jump on and off buses all day and evening.
Just pick an apartment that is on or very near a bus route.
You can download the app for Lothian Regional Transport to see where the bus routes take you, and compare that to apartment locations.

We stayed at A-Haven guest house/B&B. Easy to reach by bus from the airport. There’s a bus stop right at the corner, so it’s easy to get downtown to visit the sites. From it, one can easily walk to Leith and Royal Yacht area. We also took busses to visit Craigmiller castle and Rosslyn chapel.

Thanks again everyone. Nice to hear from a fellow Canadian as well!
Caroline