Ok, so you're a tourist, so you will most likely be trudging around all day. Hiking boots are, well, heavy in the city so instead consider the options of a trendy sneaker, biker boots or, for rainy days, a pair of Wellington boots, preferably from Hunter.
It rains, a lot, in Iceland. However, you will hardly spot Icelanders carrying an umbrella. Why? Well, it's simply totally inefficient here. The wind tends to blow sideways which means either that you get horizontal rain or your umbrella will get ruined.