Handlingen är långsam i början, men mitt nya bokmärke ligger redan mitt i boken.

Here, handlingen means the plot or the storyline of the book.

A few useful notes:

• handling = plot, story action, action, act, procedure, depending on context
• handlingen = the plot

In a sentence about a book, handlingen will usually be understood as the plot.

Because Swedish usually puts the definite article on the end of the noun.

So:

• handling = plot
• handlingen = the plot
• bok = book
• boken = the book

This is one of the biggest differences from English. Instead of a separate the, Swedish often uses a suffix.

Because långsam is an adjective describing the noun handlingen through the verb är.

In Swedish, after är, the adjective agrees with the noun:

• en-word singular: långsam
• ett-word singular: långsamt
• plural: långsamma

Since handlingen is an en-word singular noun, långsam is correct.

Compare:

• Handlingen är långsam. = The plot is slow.
• Det går långsamt. = It is going slowly.

In the second example, långsamt works more like an adverb.

I början means in the beginning or at the beginning.

It is a very common fixed expression:

• i början av boken = at the beginning of the book
• i början av filmen = at the beginning of the film

English speakers sometimes want to use a different preposition, but in Swedish this phrase normally uses i.

Because bokmärke is an ett-word.

That affects the possessive:

• min = my, for en-words
• mitt = my, for ett-words
• mina = my, for plural nouns

So:

• en bok → min bok
• ett bokmärke → mitt bokmärke

The adjective becomes nya because after a possessive, Swedish uses the weak adjective form:

• mitt nya bokmärke = my new bookmark

Bokmärke means bookmark.

It is made up of:

• bok = book
• märke = mark, sign, label

So it is very close to the English idea of a book-mark.

Swedish often uses special verbs for the position of things:

• ligga = lie
• stå = stand
• sitta = sit

A bookmark inside a book is thought of as lying, so ligger sounds natural:

• Bokmärket ligger i boken.

Using är would usually sound less idiomatic if you are talking about physical position. Swedish prefers these position verbs more than English does.

Redan means already.

In this sentence:

• mitt nya bokmärke ligger redan mitt i boken

it comes after the finite verb ligger, which is a very normal position in a main clause.

So the structure is:

• subject: mitt nya bokmärke
• verb: ligger
• sentence adverb: redan
• rest of the sentence: mitt i boken

This placement is natural and neutral.

Mitt i boken means in the middle of the book or right in the middle of the book.

The expression mitt i is very common and means right in the middle of:

• mitt i stan = right in the middle of town
• mitt i natten = in the middle of the night
• mitt i boken = right in the middle of the book

So the sentence is saying the bookmark has already reached the middle part of the book.

No. They just look the same.

In mitt nya bokmärke:

• mitt = my for an ett-word noun

In mitt i boken:

• mitt means middle or right in the middle

So these are two different words with different functions:

• mitt = my
• mitt i = in the middle of

That is why the sentence can look confusing at first.

Men means but, and it links two main clauses.

The second clause still follows normal Swedish main-clause word order, where the finite verb comes in second position:

• mitt nya bokmärke = first element
• ligger = second element

So:

• ..., men mitt nya bokmärke ligger redan mitt i boken.

This is completely normal Swedish word order. If you started the clause with another element, the verb would still stay in second position:

• ..., men redan ligger mitt nya bokmärke mitt i boken.

That version is possible, but it sounds more marked. The original sentence is the more natural neutral phrasing.