Hey Sam!
I just moved to Reykjavik in October of this year from NYC. Will you be moving to Reykjavik or elsewhere in Iceland? This information is really only relevant if you are living in Reykjavik
Make sure you bring plenty of warm clothes to layer as is more expensive here to buy thermal layers. There are plenty of second hand shops, a flea market, and a mall if you should need more but I would recommend stocking up before you come. Forget the umbrella since it's too windy here. Focus on a good coat, hat, scarf, and gloves.
Regarding language, pretty much everyone speaks English and most are willing to communicate in this language so you shouldn't have a problem there. Occasionally older generations will be a bit more stubborn in sticking with their native tongue, but who can blame them? There are plenty of classes you can take and even a degree program if you are that interested in learning Icelandic. It is a very tough language with lots of confusing rules. Personally I just pick things up from my friends as needed and use the phrases I learn as much as possible. Icelanders really appreciate an attempt to learn their language.
Cultural differences....let's see...I'd say it's really easy to meet and interact with Icelandic people but sometimes a bit hard to be included in their inner circle (ie: being invited to go out or get a coffee). I was lucky enough to come here quite a few times and make friends before moving. Lots of people say they end up becoming better friends with foreigners. So try taking a class and don't be afraid to go out on your own to meet Icelanders. They are a lovely bunch And be patient. It'll work out!
That's the basics I can think of for now. I hope that helps!