I live in the city centre of Edinburgh, and can pretty much walk most places within 20 minutes, and can get to the opposite side of town in 40 minutes. I had a bus pass for the first year and a half I lived here, but abandoned it in favour of walking because Genau is right about Edinburgh public transport...the buses are pretty rubbish. BUT, they do get you to where you want to go in most cases...there are very few places in the city that are unreachable by public transport. I probably take the bus once every week or two. It's extremely walkable. And hopefully all the faff with the trams will be worth it in the end!
I lived in London for four years, and took a bus/tube combo every day. I would NEVER ever EVER give up my laid-back pedestrian Edinburgh lifestyle for that circus ever again!