Ah, this is my field! Brace yourself TheJeff, a wall of text is coming:
When precisely are you planning on arriving? If you're arriving between August 3rd - 28th, there is an absolutely ENORMOUS arts festival called the "Fringe". I mean so vast that it entirely takes over the center of the city with a carnival-like atmosphere, and there are literally thousands of music gigs, theatre productions, stand-up comedy shows, "alternative" events, etc. Almost all are cheaper than £15 (not sure what that is in dollars), and many are free (although it's considered very bad courtesy not to give something at the end, I'd say worse than not tipping a waiter in the US).
Now, whether this is a good thing or not depends entirely on your appetite for that type of stuff. Personally, I love it, and performing in, and experiencing, the Fringe is actually a huge reason I came here. However, if it's not your scene, or if you aren't a fan of crowds, it's best avoiding the center altogether throughout this time. With regard to running, the Meadows is basically impossible to run on during the Fringe, it would be like trying to run through a mosh pit.
Now, tourist stuff! I work as a tour guide, so this is my jam. There's the Edinburgh ghost tours, which can genuinely be an interesting way to see a bit of the city, and could be a huge amount of fun (or totally terrifying) for the children. However, get a walking tour rather than one from a bus. You can also do straight history tours or walking tours. On a similar note, there's the Edinburgh Dungeons, which can be fun for an immersive Middle Ages Scottish horror themed hour. Both the Dungeons and the tours are typically led by very good guides, as opposed to a tourist rip-off (mainly because Edinburgh has a wonderful acting scene, and a lot of the participants moonlight as guides). However, if you're here during the Fringe, avoid doing a walking tour during the day. It ends up as an elongated fight against people-traffic.
As well as a good running route, Arthur's Seat is an unchallenging walk that offers great views of the entire city. There's also the Pentlands, and if you're a hiking fan, it's definitely worth going to Loch Lomond for a day, or even further north to the Highlands for a little longer.
The Castle (at the top of the Royal Mile) is a little expensive to enter, but I think it's worth it. It's a great little snapshot of history.
There is also the Scottish Parliament building at the other end of the Royal Mile, which is open to the public, and you can sit in on live debates. Also, in the 18th century, the 4-minute mile was broken by men running down the Royal Mile from the Castle to where the parliament now stands (it is, obviously, a downhill route).
There's also a number of old cemeteries (Canon's Gait Kirkyard, Greyfriars Kirkyard, and the Old Calton burial ground) in which there's a number of cool-looking monuments and stories of historical figures such as the economist Adam Smith, or the philosopher David Hume. If anyone had children into Harry Potter, there's the names of several of the characters on the graves in Greyfriars - JK Rowling wrote the first novel in Edinburgh, and took their names. On a related note, there's Harry Potter tours also running throughout the year.
I could type all day on this topic, but I don't want to wear my fingers to stubs, and I have work to do. However, if anyone has any questions about life in Edinburgh, I'd be more than happy to answer. I hope you enjoy your trip!