I know it’s probably obvious to people who’ve lived in the city for a long time, but having just got here (from London…) and explored a bit, I think it’s really useful to emphasise how great a place Edinburgh is in terms of things to do, even for free, and particularly for those unused to exploring a smaller, more European feeling city by foot. The particular difference between the two capitals is that Edinburgh has the same calibre of stuff that London does, just scaled down and mostly all a Tube-less, pleasant, scenic walk away.
First, the really obvious first two amazing things: Edinburgh has a castle and a (almost) mountain. You know, just in the middle of the city. To get in to the castle is, admittedly, a bit pricey (http://www.edinburghcastle.gov.uk/tickets/tickets.aspx), but just having it there as you have a tourist day and walk around Grassmarket or the Royal Mile is great. The aforementioned mountain is completely free to just walk over to, on, up and over. Be careful not to get blown off the top though.
What else? A short walk from the Royal Mile is the National Museum. This is much smaller than its London counterpart, but is in many ways much slicker, maybe because of its size. It’s got natural history, some great stuff about technology, and lots of interactive exhibits for childish adults and actual kids.
In the less visited (at least by tourists) south of the city, there’s also The Meadows, a wide, tree lined park area that’s a favourite for barbecues during the summer. Did I mention there’s also a beach? Yep. A short bus ride from the city centre is Portobello which, though a bit run down, has about a mile’s worth of sand, fish and chip shops, and ice cream parlours. Both of these places are lovely during Edinburgh’s intermittent sunny periods.
Lastly, we of course shouldn’t forget to mention Edinburgh’s vibrant night life, beautiful, small, music-filled pubs, and the way all of them really come to life during the city festival during the summer.