Just behind Hoolyrood Palace is a nice park and a series of “mountains” and cliffs. The highest of these is called Authur’s Seat. We set out from our hotel, walked 1/2 mile down the Royal Mile (downhill) to the park.
If you are smart, you would stop at the information center and get a map of the trails to the top. We stopped here after we descended so we could see where we had gone.