I agree with madchen about Sneakys and Ventoux. The Jazz Bar for live jazz, Sorra Lella does amazing italian plant based food, Central Mosque Kitchen is cheap and delicious. Edi is the only city in Scotland u can legally drink on the street so if the weather is nice head to The Meadows. Wandering about the city is good but its so easy/ cheap to get out to some more rural green spaces, Rosslyn Chapel and woodland etc or Pentlands if ur into hikes.
Jorum studio is niche perfumers in Stockbridge, Armstrongs is famous chain of vintage shops (expect a search for something good), charity shops in Stockbridge and South Bridge are better priced and more variety imo, 5th season vintage does more curated /90s european designer vintage if thats ur thing
There's also a cast iron fountain on Castle Hill dedicated to victims of the witch hunt which I always take people to see