Calton Hill is fine during the day and a nice viewpoint along Princes Street.
Wouldn't go there after dark though, well, not unless I wanted some manlove with a stranger and that desire has passed me by so far 🙂
Can walk from there to Holyrood, up Arthur's Seat and Salisbury Crags and out by the Commie Pool into Newington and back across Meadows into the city. Once you get out the student bit there's plenty of nice cafes and so on in Marchmont/ Bruntsfield.
Depends what you're after really. Meant to be a nice day anway.
If you're up in late morning/ lunchtime, Farmer's Market on Castle Terrace is worth a look as well (opposite end of the city centre to Calton Hill though).