Welcome, Flykuni!
I'm an Edinburgh man myself. During the festival, the city is pretty busy - hope you've got your accommodation sorted out. So long as you don't mind the odd drunk Scotsman (some, very odd) then you'll have a blast. Let me rephrase.
Everybody will be drunk, but only a few of them will need to be avoided. Most of the locals are actually reasonably friendly. They won't go out their way to put on a big show of welcome, but we're not insincere either. We get loads of visitors from round the world so one more American is not going to have the tumble weed blowing while the honky tonk from the piano in the corner fades away. And in the unlikely event that you do get any bother, just say you're Canadian.
Do be prepared for service that is somewhat less swift than you might expect on your home soil. But keep an open mind, enjoy the differences, and you'll be sorted.
The Water of Leith runs within a couple of hundred yards of the city centre (the New Town part of it at least, "New" being Georgian ie 1780ish if my memory serves). It's reputed to have a decent stock of wee wild brownies in it. It's a nice walk from the town centre upstream to the Modern Art Gallery (free entry there too). There's a thread on here somewhere with loads of info about fishing the Water of Leith. I'm a loch man, myself, so afraid I can't offer any help on the rivers.
Oh, and I know it sounds touristy and cheesy, but starting your visit with an open top bus tour is a great way to get your bearings. Or climb Arthur's Seat and look over the city.
Other tips? Bring comfy walking shoes. Bring a light raincoat. Edinburgh Castle is magnificent from the outside, but overpriced to get in. And going up Calton Hill at night can be a bit dodgy. So I'm told.
Neil