Well, I would say knowing when to use the definite articles are important, but it won't jeopardise the meaning you want to carry across (perhaps with some exceptions). As Pierre-Emmanuel mentioned, these definite articles are placed as suffixes, although there are others (sá, sú, það) that would mean the same thing, but placed in front of the word. For example: sá hundur (the dog), sú bók (the book), það hús (the house), these though have a secondary function that won't be important until later stages.
It might be good to know that these definite suffixes also decline into the four cases, i.e. hundurinn, um hundinn, frá hundinum, til hundsins (the dog, about the dog, from the dog, to the dog). Of course, if you get this wrong, you won't be misunderstood, but it's always good to speak/write as grammatically correct as you can. =) Icelanders make mistakes too.
"Að" has any functions, and it'll be good to know them by context. I.e. "að' as an infinitive marker (að hlaupa 'to run'), or as a preposition 'towards' (að húsinu 'towards the house').
As far as Icelandic grammar is concerned, don't worry about having to learn or memorise rules in the beginning, but it's good to know of them and take note of their usage. From my experience. =) Hope it helps!