Dictionaries
Orðabók - this online dictionary is subscription only, but is my favourite way to look up words. It works great on smartphones, and provides long and detailed entries for most words with many definitions of phrases and alternate meanings. It will also usually correctly identify the headword from whatever inflection of a word you might type in. At about €12 a year for one subscription it's excellent value for money.
BÍN (Beygingarlýsing íslensks nútímamáls) is an amazing collection of all the words in Icelandic in every case and inflection possible. If you want to work out how to decline a word or find the root infinitive and so on, this is where to go.
University of Winsonsin-Madison has a good free online dictionary. It's not quite as good as Orðabók but on occasion it has better entries with idioms and usage.
Íslensk-ensk Ensk-íslensk Vasaorðabók (Forlagið) This is my favourite physical dictionary - it's a bulky pocket-sized book but usually contains whatever Icelandic word you're looking for.
Íslensk Orðabók is recommended for intermediate and advanced learners. It's very comprehensive but entirely in Icelandic with no English. I wouldn't use it all the time but it's invaluable for those moments when you just can't find a word or definition in other sources.
Icelandic-English English-Icelandic Practical Dictionary is highly recommended for beginners, as it is a fairly practical size and is very user-friendly. However once you progress to an intermediate level you'll find more and more that the word you're looking up isn't there.
Íslensk barnaorðabók is not a dictionary you would use to look up words, rather a fun book to peruse and read definitions in Icelandic of some 2,200 headwords. Good as a way to start leaving bilingual dictionaries behind.
Textbooks
Íslenska fyrir alla (Icelandic for all) is my favourite text, and is often used by people who have moved to Iceland. It's entirely in Icelandic without any English explanations, but with a good dictionary at hand it is easy to follow and the learning curve is gentle. There are excellent quality audio clips available for free online. The emphasis is on ordinary speech in everyday situations, and by the time you finish Book 4 you will be really quite advanced in your knowledge of the language.
Beginner's Icelandic is a good text, although I wouldn't recommend it for complete beginners. It has lots of dialogues, most of which are on the accompanying CDs (what's a CD I hear many of you cry), and good explanations in English of various grammar points. The part on hesitation words (erm etc.) is particularly useful.
Complete Icelandic: Teach Yourself is perhaps the best book in English for beginners. It also comes in a digital version with embedded audio.
Colloquial Icelandic is also a very good book, but I think the learning curve in it is a bit extreme. By unit 2 it's already throwing words like 'attaché' and 'marketing director' at you. But like Complete Icelandic, it covers a lot of ground in one volume.
Icelandic by Stefán Einarsson is a really old book written during WW1, intended for US Army officers destined for duty in Iceland. Nevertheless it is a very interesting text with extremely detailed explanations of Icelandic grammar and lots of graded reading exercises.
Learning Icelandic is a curious pair of books. The main book contains 15 chapters of lessons with dialogues (audio available for free online) with new vocabulary listed at the bottom of each page. It also has a very good but challenging explanation of Icelandic grammar in English, which is good for getting into cases and verb inflections. The second book contains lots of grammar exercises.
Íslenska fyrir útlendingur: Kennslubók á málfræði is a very good book for when you feel ready to talk about Icelandic grammar entirely in Icelandic. There are many useful lists and tables of verb forms and so on and lots of example phrases. There is a separate exercise book as well. This is more of a reference book than a book to use as a simple learning course.
Other Resources
Icelandic Online is an excellent free course provided by the University of Iceland. It used to be Flash-only but they have converted almost all of it to HTML5 now. It will take you from beginner to advanced and has many good interactive features, but is more and more weighted towards literature and poetry as you progress.
Storytel is a Swedish audiobook company with a site dedicated to Icelandic audiobooks. For a monthly subscription of about €20 you can access a large library of Icelandic audio.
Icelandic Cinema Online has a collection of 180 Icelandic movies available for streaming. Most movies cost €3 or €4 for 24 hours of access.
Kindle is a good source of Icelandic e-books, many of which are expensive and difficult to find outside Iceland in actual print. Usually it's best when you know what you are looking for, such as kids' books by Gunnar Helgason or grown-up books by stand-up comedian Jón Gnarr.
Books in Icelandic are often hard to find outside Iceland, and tend to be expensive in the country. Whenever I visit I try to stock up on as many books as I can. If you travel there and are looking for common and popular books, be aware that you might find them 10% cheaper at the airport duty free bookstore due to the ridiculous book sales tax. (You can peruse the airport bookshop when you arrive but can only buy when you are flying out.) You can order books online but shipping is very expensive (€25+ per order). The best value is usually found at the University Bookshop Bóksala Stúdenta but you can also order online from Forlagið and
Eymundsson.
AudioStretch is one of many useful smartphone apps you can enlist to help you in your studies. This particular app lets you import an mp3 file and manipulate it in various ways, such as slowing it down, looping a section or even scrubbing the waveform with your finger.
RÚV (the national broadcaster) has a good app for watching Icelandic TV. Some shows are geoblocked outside Iceland, such as movies and dramas, but most factual content is available. You can also listen to the radio.
Surprisingly the Strætó app for catching the bus in Reykjavík also has a useful podcast feature. Download the app to your phone to listen to an interesting curation of Icelandic podcasts while on the bus or sitting at home.
The Icelandic Summer Course takes place at the University of Iceland each July. If you are keen about learning Icelandic I can't recommend it highly enough. The teaching at three different levels is superb and it's a great way to spend a month in the capital during the endless days of summer.