We're going to Iceland for the first time in September. Will we be able to get around ok just speaking English, or should we try to learn some rudimentary Icelandic? And is that even what their language is called?
Yes, most Icelandic people speak very good English, you will be fine.
Their language is indeed called Icelandic, and is fascinating in regards to its history. From wikipedia:
The oldest preserved texts in Icelandic were written around 1100AD. The majority of these texts are poems or laws, preserved orally for generations before being written down. The most famous of these, written in Iceland from the 12th century onward, are without doubt the Icelandic Sagas, the historical writings of Snorri Sturluson; and eddaic poems.
With regards to learning a few Icelandic phrases, as with anywhere, people appreciate non-native speakers making the effort. "Takk" for "Thanks" is an easy one for starters. Some of the pronunciation is nigh on impossible for English speakers who haven't been exposed to it before, so don't think you'll attain even a basic grasp without a lot of study!
An Icelander will usually start speaking English with you before you have a chance to open your mouth. This is very disappointing to people like me that go to Iceland to learn the language. Other students I know will resort to wearing a sign around their neck that begs people to stick to Icelandic.
The language name in their own language is Íslenska. So you'd say (or write on a card) Ég tala ekki íslensku. Which will be obvious as soon as you open your mouth :-)
Better than learning phrases--unless you are a foreign language hobbyist, I'd learn other factoids, such as the emergency number is 112.
Enjoy your trip.