Ég geri það mikilvægasta fyrst.

Gera is the dictionary form, meaning to do / to make.
In the sentence, the verb has to match ég (I), so it becomes geri, the 1st person singular present tense.

A few present-tense forms of gera:

• ég geri — I do
• þú gerir — you do
• hann/hún/það gerir — he/she/it does
• við gerum — we do
• þið gerið — you (plural) do
• þeir/þær/þau gera — they do

So Ég geri ... simply means I do ...

Here það does not feel as strongly like English that as it often does in other contexts.

In það mikilvægasta, it helps create a noun-like phrase meaning:

• the most important thing
• what is most important
• the most important one

So the whole phrase það mikilvægasta works as the object of the verb.

Icelandic usually does not need a separate word like English most.
Instead, the adjective itself changes form.

For mikilvægur (important):

• mikilvægur — important
• mikilvægari — more important
• mikilvægastur — most important

So the idea of most is built into mikilvægasta.

Because the adjective has to match the phrase it belongs to.

In það mikilvægasta:

• það is neuter singular
• the adjective is being used in a definite or noun-like phrase
• so the superlative adjective appears in the matching form mikilvægasta

Very roughly:

• mikilvægastur = masculine form
• mikilvægast = neuter, often more indefinite
• mikilvægasta = the form used here with það, meaning the most important thing / the most important one

So this is an agreement issue: the adjective changes to fit the grammar of the phrase.

Yes, you can think of there being an understood noun such as:

• thing
• task
• part
• one

Icelandic often allows adjectives to stand on their own when the noun is understood.

So það mikilvægasta literally feels like:

• the most important [thing]
• what is most important

English often adds a noun where Icelandic does not need one.

It is the accusative, because it is the direct object of geri:

• Ég geri hvað? — það mikilvægasta

However, in the neuter singular, the nominative and accusative often look the same, so you do not see an obvious ending change here.

Fyrst is an adverb meaning first. In Icelandic, adverbs like this often come later in the sentence, and putting it at the end is very natural here:

• Ég geri það mikilvægasta fyrst.

That gives the sense:

• I do the most important thing first.

You can move things around for emphasis, but the sentence you were given is a very normal, neutral way to say it.

For example:

• Fyrst geri ég það mikilvægasta. = First, I do the most important thing.

Notice that when fyrst is moved to the front, the finite verb geri still stays in second position. That is a very important Icelandic word-order pattern.

Here it means first in order:

• I do the most important thing first.

That is different from English at first, which usually means initially, but later something changed. Icelandic often expresses that idea differently, for example with í fyrstu in the right context.

So in this sentence, fyrst is about sequence, not about a change over time and not just a formal list word like firstly.