Learn Hardcore Icelandic: Ég fer í búð til að kaupa mjólk.

fara is the infinitive meaning to go. In Icelandic you must conjugate it for person and number in the present tense, so Ég fer is the 1st person singular present form of fara.

Why is there no article before búð? Wouldn't the store be búðin?

In this sentence you’re speaking of going shopping in general, not a specific store. Indefinite singular nouns remain unmarked: búð = a shop/store. If you wanted to say the store, you would use the definite form búðin.

Why is í used with búð, and why does búð have no visible case ending?

The preposition í can govern either the accusative (for motion toward) or the dative (for location).
Here you’re expressing motion into the shop, so í takes the accusative case.
Many feminine nouns (like búð) have identical nominative and accusative singular forms, so there’s no extra ending to show it.

What is the function of til að here?

til að is a fixed purpose marker equivalent to English in order to or the infinitival to when expressing purpose. The word til by itself usually governs a noun in the genitive (e.g. til vors = until spring), but when you want to attach a verb for purpose, you always follow til with að plus the infinitive.

Could you drop til and say Ég fer að kaupa mjólk?

No. Ég fer að kaupa mjólk would be misinterpreted as an attempt at the near-future or progressive construction (I’m going to/starting to buy milk), and it still needs er if you use that pattern (Ég er að fara að kaupa mjólk). To express purpose after fer, you must use til að.

What case is mjólk, and why does it look like the nominative form?

mjólk here is the accusative singular object of kaupa. Many feminine nouns have identical nominative and accusative singular spellings, so you don’t see any change. You know it’s the object because it follows the verb kaupa, which takes a direct object in the accusative.

How would I pronounce í, ú and ð in this sentence?

í sounds like the ee in see (tenser).
ú sounds like the oo in food.
ð is like the voiced th in this.

So búð ≈ booth, mjólk ≈ myoulk, and the whole sentence approximates yegh fehr ee booth til ath koy-pa myoulk.