The official language of Iceland is Icelandic. English is very widely spoken, and it's unlikely that you'll encounter anyone in your day-to-day life who isn't able to communicate effectively in English.
However, it will certainly smooth your daily interactions to understand key phrases used in shops and restaurants. It will also be helpful to be able to say your phone number and kennitala in Icelandic.
There are a number of institutions offering Icelandic courses aimed at foreigners. Most of them are heavily subsidised by your union membership.