This is a list of the grammar you learn in the Bachelor’s Degree Icelandic for Foreigners course at the University of Iceland, in the first semester. If you take the degree course you’re basically expected to know this stuff already or you’re expected to be able to learn it in two days to a week. I’ve explained the rules for some things that might be harder to find elsewhere, but if you want me to explain anything in this post (or explain it better), then just ask. I’m not saying my grammar explanations are perfect, I’ve seen my textbooks and teachers have mistakes and I might have made mistakes when I was copying the info down.

(All cases for everything were covered but past tense wasn’t covered.)

Gender of nouns and adjectives
Definite articles
Personal pronouns
Já vs. jú usage
Possessive pronouns
Past tense (briefly but not completely?)
Imperative (commands)
Using heiman/heima/heim, ofan/uppi/upp, neðan/niðri/niður, innan/inni/inn, utan/úti/út (coming from/no movement is happening/leaving)
Direction words
Ordinal (dates) and cardinal (regular) numbers
Who/what/when/where/why question words (all cases)

Þurfa vs. verða: “must/need”
Verða is “stronger”.
Þú þarft að kaupa mjólk. Það er lítið til af mjólk í ísskápnum. You have to buy milk. There’s little left of the milk in the fridge.
Ég verð að kaupa í matinn fyrir kl. 6. Það er enginn matur til í húsinu. I need to buy food for six o’clock. There’s no food left in the house.

Eiga vs. skulu: “shall”.
Skulu is never used in questions, is used for asking advice, giving suggestions, and is used when offering/promising.
Eiga is used in questions and instructions.

Eiga vs. hafa vs. vera með: to own/have/be with
Eiga is a thing owned by the subject of the sentence (ég á þessa bók – I own this book), or it indicates social relationship (þau eiga tvö börn – they have two children).

Hafa is used for owning something abstract (not physical, ég hef ekki tíma núna – I don’t have time now) and is used like “I have read this book” just like in English.

Vera með is used for body parts, illnesses, clothes, accessories (scarves etc.), or if the object of the sentence is something the subject is carrying temporarily (Anna er með bókina – Anna has the book [with her right now]).

Ég á bókina en ég er ekki með hana núna. I own the book but I don’t have it with me now.
Ég á enga peninga. I don’t have any money/I’m broke.
Ég er ekki með peninga. I don’t have any money on me.

J-innskot (J-insertion):
If a stem ends in ý, æ, ey, g, or k there’s a j added between the stem and the ending in singular plural and third person plural in the present tense.
Examples: flýja, deyja, hlæja, leggja, hringja, syngja.

If the stem ends on other sounds then a j is inserted in all plural forms in the present.
Examples: telja, spyrja, sitja

If the stem ends in ý, æ, or ey and the ending starts with a or u, then a j is inserted.
Examples: nýr, bær, Laufey

Exceptions: tala, heyra, koma, taka, auka

A-vixl: (A-mutation)
In the stem of a word, a changes to ö if the ending starts on the vowel u or if there’s no ending. If the previous vowel in a stem is a but the second isn’t, a-vixl doesn’t happen. If there are two vowels and the second vowel is an a, it changes to u instead.
Examples: Ég tala – við tölum. Barn – börn. Hundrað – Hundruð.

B-vixl:
The B-vixl encompasses many vowel changes instead of just one like with the A-vixl. It’s in the present singular in strong verbs, never in weak verbs.

A to e: fara (ég fer)
ö to e: stökkva (ég stekk)
o to e: koma (ég kem)

o to æ: þvo (ég þvæ)
ó to æ: róa (ég ræ)
á to æ: láta (ég læt)

ú to ý: súpa (ég sýp)
jú to ý: fljúga (ég flýg)
jó to ý: bjóða (ég býð)

au to ey: auka (ég eyk)
já to é: sjá (ég sé) – Remember that é isn’t just é, it’s actually je (and that’s still the sound it makes even if it’s spelled differently). That might help you to remember.

C-víxl:
Happens in the past and with past participles, with strong verbs. Strong verbs that have four mismatching stem-changes have the C-vixl. (This is all the C-vixl is to my knowledge, there’s no real pattern to it that I can list.)