Hi there,
I realize you've probably already started your program given the date of your post but I thought this might help future students. I did my Master's at Edinburgh and it's a fantastic city. I lived on the edge of Stockbridge and New Town. It's a very nice area but kind of a long walk to the uni (40 minutes give or take) but there are a few different buses that drop you off right at the university. It was about a 15 minute walk to Princes Street, which is the central road that divides New Town from Old Town. I never had a car and never had much use for one as the bus system is very good. The University is in Old Town and it's generally cheaper rent-wise, but there are some parts that aren't very nice. If it's financially viable, I would avoid the Leith area. There are some nicer parts (The Shore is up-and-coming and Ocean Terminal is a brand new mall out there) but there are some very bad neighborhoods and it's quite far from the uni.
I wouldn't bring any furniture or big ticket items with you unless you just can't live without them. There are lots of fully furnished flats available (that includes everything from beds, a sofa and a washing machine to cutlery, plates and glasses) and shipping is insanely expensive. In my opinion it's not worth it even if you will be living there for a number of years.
I'm not really sure about Americans renting flats there because I'm married to a Brit who sorted it all out, but I imagine paying a few months up front would quell any landlord's fears. Most of the international students lived in grad housing but I don't know what's available for couples. Just to give you an idea of expenses, three years ago, our extremely old but spacious two-bedroom flat in a very nice neighborhood cost 600 pounds a month, fully furnished. That was a pretty good deal at the time, and most of the flats in Stockbridge were about that for one bedroom. We were pretty careful with money and spent about 50 pounds per week on food/household goods for the two of us. The phone/internet bill was paid every three months and came to about 100 pounds.
I hope that helps