Ég tek hjólið úr hjólageymslunni.

Taka is the dictionary form, meaning to take.
In the sentence, you need the present tense, first person singular form: ég tek = I take / I am taking.

So:

• taka = to take
• ég tek = I take

This verb is not fully regular, so the vowel changes from a to e.

It can mean either, depending on context.

Icelandic often uses the simple present where English might use either:

• I take
• I am taking

So Ég tek hjólið úr hjólageymslunni could mean:

• I take the bike out of the bike storage room
  or
• I am taking the bike out of the bike storage room

The situation usually makes it clear.

The ending -ið is the definite article attached to the noun. Icelandic usually adds the to the end of the word instead of using a separate word.

So:

• hjól = bike / wheel
• hjólið = the bike / the wheel

That is why there is no separate word for the in the sentence.

Because it is the direct object of the verb taka.

In Icelandic, taka normally takes the accusative case, so the thing being taken goes into the accusative.

Here, hjólið is the thing being taken, so it is in the accusative. With hjól, the nominative and accusative singular definite happen to look the same, so you do not see a visible difference here.

Úr takes the dative case.

So the noun after úr must be in the dative:

• úr hjólageymslunni = out of / from the bike storage room

This is something learners usually need to memorize with prepositions:
úr + dative

Because úr requires the dative, and hjólageymslunni is the dative singular definite form.

Compare:

• hjólageymsla = a bike storage room
• hjólageymslan = the bike storage room (nominative)
• hjólageymsluna = the bike storage room (accusative)
• hjólageymslunni = the bike storage room (dative)

Since the sentence uses úr, the correct form is hjólageymslunni.

It is a compound noun:

• hjóla- = of bikes / bicycles
• geymsla = storage, storage room

So hjólageymsla means something like bike storage, bicycle storage room, or bike shed, depending on context.

Compound nouns are very common in Icelandic, just as in English.

Yes. Hjól can mean wheel, but in everyday Icelandic it can also refer to a bicycle in the right context.

So the exact meaning depends on the situation. In a sentence with hjólageymsla (bike storage), hjólið is naturally understood as the bike.

Because Icelandic úr already covers the idea of out of / from inside.

So:

• úr húsinu = out of the house
• úr hjólageymslunni = out of the bike storage room

Sometimes Icelandic also uses út úr for extra emphasis, but úr by itself is completely normal here.

Both can often be translated as from, but they are not the same.

• úr usually means out of, from inside
• frá means from, often more generally, not necessarily from inside something

So here úr hjólageymslunni is the natural choice because the bike is being taken out of a storage space.

Yes, but the given order is the most neutral one.

• Ég tek hjólið úr hjólageymslunni. = neutral, ordinary statement

Icelandic word order can change for emphasis or style, but the verb usually stays in the second position in main clauses. For example:

• Á hverjum morgni tek ég hjólið úr hjólageymslunni.
  Every morning I take the bike out of the bike storage room.

So word order is somewhat flexible, but not random.

The hj combination does not sound like a strong English h followed by j. It is pronounced more like a soft hy sound.

A rough English-friendly approximation of hjólið is:

HYO-lið
with the ð sounding like the th in this

The exact Icelandic sound is a bit different from English, but thinking of hj as a soft hy is a useful start for learners.