Á kvöldin fer ég í náttfötin og les bók í rúminu.

Because Icelandic main clauses normally follow the verb-second rule.

If you put a time expression like Á kvöldin at the beginning, the finite verb still has to come second, so you get:

Á kvöldin fer ég ...

not

Á kvöldin ég fer ...

A more neutral order would be:

Ég fer í náttfötin á kvöldin ...

Both are possible, but the original sentence starts with the time phrase for emphasis or topic.

Á kvöldin means in the evenings or every evening.

This is an idiomatic Icelandic time expression. Even though á often means on, in expressions like this it is simply the normal way to talk about repeated times of day:

• á morgnana = in the mornings
• á kvöldin = in the evenings
• á næturnar = at night / during the nights

So you should learn á kvöldin as a set expression meaning a habitual time.

Here, kvöldin is the definite plural form of kvöld.

In the expression á kvöldin, Icelandic uses this form idiomatically to mean in the evenings as a repeated habit. So although it looks like the evenings, the whole phrase functions more like an adverbial expression of time than a literal noun phrase.

This is one of those patterns that is best learned as a chunk: á kvöldin = in the evenings.

Yes, fara usually means to go, but in Icelandic it is also used in expressions like fara í föt meaning to get into clothes / put clothes on.

So:

fara í náttfötin
means
to get into the pajamas / to put on pajamas

This is very natural Icelandic.

A useful contrast is:

• fara í náttfötin = put on / get into pajamas
• vera í náttfötum = be wearing pajamas

So fara emphasizes the change into that state.

Because í can take different cases depending on meaning:

• accusative = motion/change into
• dative = location in

So here:

• í náttfötin = into the pajamas → motion/change, so accusative
• í rúminu = in the bed → location, so dative

This is a very important Icelandic pattern:

• Ég fer í húsið. = I go into the house.
• Ég er í húsinu. = I am in the house.

The same idea is used in your sentence.

Because náttföt is normally a plural noun in Icelandic, much like pajamas or clothes in English.

So:

• náttföt = pajamas / nightclothes
• náttfötin = the pajamas

Even when English speakers think of it as one item of clothing, Icelandic usually treats it as plural.

Yes, literally it is the pajamas, because Icelandic attaches the definite article to the end of the noun:

• náttföt = pajamas
• náttfötin = the pajamas

But with clothing, Icelandic often uses the definite form where English would naturally use a possessive such as my or his, if ownership is obvious from context.

So fer ég í náttfötin can very naturally mean:

I put on my pajamas

even though the Icelandic form is literally the pajamas.

Because Icelandic does not have an indefinite article like English a/an.

So:

• bók can mean a book
• bókin means the book

That means:

les bók = reads a book

If the sentence meant reads the book, it would be:

les bókina

So the absence of a separate word for a is completely normal in Icelandic.

Because Icelandic often leaves out a repeated subject in coordinated clauses when it is obvious.

So:

Á kvöldin fer ég í náttfötin og les bók í rúminu.

literally works like:

In the evenings, I put on my pajamas and read a book in bed.

English does the same thing. We do not have to say:

... and I read a book in bed

unless we want extra emphasis.

You could say og ég les bók, but leaving out ég here is very natural.

Because Icelandic, like English, uses the present tense for habitual actions.

The phrase Á kvöldin already tells you this is something that happens regularly, so the present tense means:

In the evenings, I put on my pajamas and read a book in bed.

It does not mean only what is happening right now. It describes a routine.