Beginner's Icelandic review
I think Beginner's Icelandic is a better way to get started with Icelandic than Teach Yourself's Complete Icelandic - but even though I worked through all the exercises, I was usually reviewing grammar I'd already learned from Complete Icelandic. So maybe I've overlooked gaps in what Beginner's Icelandic covers, or unclear explanations, because it was material I'd already learned.
That said. I liked the way Beginner's Icelandic approached Icelandic grammar. I've mentioned before how Icelandic words take a huge variety of endings, and how Complete Icelandic expects you to learn all of them at once. In contrast, Beginner's Icelandic spreads the same material out across several chapters, so you practice the nominative endings first, then add the accusative endings, and so on.
Furthermore, I especially appreciate Beginner's Icelandic's explanations of the little hard-to-translate words that indicate surprise, or hesitation, or emphasis, and of the difference between three expressions that can be translated as "to have". (And the dialogues can be entertaining.)
One small annoyance: some of the words in the dialogues aren't in the vocabulary lists, so those of us who study the vocabulary lists first will still find ourselves reading fluidly along and then having to look up a new word. (At least the book puts English translations of the dialogue on the facing page, so you don't have to riffle to the back.)
Overall, if you want to learn Icelandic on your own - and you want an English explanation of what you're looking at - I'd suggest starting with Beginner's Icelandic. You could add Complete Icelandic for more practice, but you might consider the University of Iceland's free online courses instead.