Do some initial research on the language to find out what the grammatical structure is like and how it differs from your first language. If you speak English as a first language, you may be surprised to find out that Icelandic is somewhat similar. This is because both English and Icelandic are part of the Germanic family of languages. This means that the basic vocabulary of Icelandic may resemble English in many ways. For example, the word for "house" is "hús" (hoos) in Icelandic. There are many words that resemble each other in Icelandic and English, and these are called cognates. The more cognates there are in the foreign language you are learning, the easier it is to learn generally.
If you're learning Icelandic on your own, find the appropriate resources. This may be a combination of books and/or CDs as well as some helpful websites where you can learn what the language sounds like and how it is spoken (intonation, phrasing, etc.). Stay away from short phrase books as these are only useful in teaching the basic necessities when traveling in the country where the language is spoken. It is better to get a deeper understanding of how the structure of the language functions, so this means you need to learn about everything from how verbs work to declining nouns (unfortunately, Icelandic nouns do decline but are fairly straightforward). Finding the right resources will help you learn about every aspect of the language.
Start practicing! Yes, even with a basic understanding of the language you can start practicing with others who speak the language, most importantly native speakers. For a language like Icelandic, it may be difficult at first to find someone who speaks Icelandic fluently. The best places to look for native speakers include university campuses and large urban areas. If you live near a major university or a big city, start looking around for international expat groups in your area. People who know that you are learning their language are usually more than willing to share some time with you to help you learn it. People from Iceland are no exception, and are usually thrilled to learn that someone is trying to learn Icelandic.
To keep your language skills polished, start watching TV and listening to music in the language you're learning. The best place to do this is the internet. Most countries have some type of news website where you can watch videos of the news in the language. Iceland has a few such websites, including www.ruv.is where you can view the latest news out of Iceland in Icelandic. Do some more research online and found out what the most popular artists and bands are in the country you're interested in. Pop music is usually easier to understand than more traditional genres of music, so start there.