Jag vill inte bara läsa boken, utan också skriva några rader i min dagbok.

In Swedish main clauses, the finite verb usually comes in the second position (the V2 rule). Here the finite verb is vill (present tense of vilja), so it stays early: Jag vill ....
Negation (inte) typically comes after the finite verb in a main clause: Jag vill inte ....
You can get inte earlier, but then you’re usually fronting something else and still keeping V2, e.g. Inte vill jag bara läsa... (more marked/emphatic).

After modal verbs like vilja (want), kunna (can), ska (will/shall), måste (must), Swedish normally uses a bare infinitive (no att):

• Jag vill läsa (not Jag vill att läsa)
• Jag vill skriva
  So läsa and skriva are infinitives directly dependent on vill.

This is a common paired construction meaning “not only ..., but also ...”. In Swedish it’s typically:

• inte bara X, utan också Y

A few notes:

• utan here means “but/rather” (not “without”).
• också = “also/too” reinforces the second part.
• You’ll also see inte bara ..., utan ... (dropping också) in less formal or less symmetrical phrasing, but utan också is very common and clear.

Swedish often uses a comma to separate the two parts of the paired structure inte bara ..., utan också ..., especially when each side is a fairly complete phrase (here, two infinitive phrases).
It’s not strictly required in every short sentence, but it’s very standard and helps readability.

Boken is the definite form (“the book”), which implies a specific book known in context. Swedish often encodes definiteness directly on the noun with a suffix:

• bok = book
• en bok = a book
• boken = the book

So läsa boken suggests “read the book (that we’re talking about)”.

Bok is an en-word (common gender), so its definite singular suffix is typically -en:

• en bok → boken

For ett-words (neuter), the definite suffix is usually -et, e.g. ett hus → huset.

Några means “a few / some” and is used with plural nouns:

• några rader = a few lines

Någon is singular (common gender) and means “some / any / someone” depending on context:

• någon rad = some line / any line (one line)

So the plural noun rader requires några.

Rad (line/row) forms the indefinite plural as rader (a common plural pattern in Swedish):

• en rad → rader

Pronunciation (rough guide): rader is typically two syllables, something like RAH-der (with a Swedish “r”, which varies by region).

With writing in a diary, Swedish normally conceptualizes the writing as being “in” the diary, so i is standard:

• skriva i min dagbok = write in my diary

På is used more for “on (a surface)” or certain fixed phrases (e.g. på papper, “on paper”), but for diaries/notebooks the idiomatic preposition is usually i.

Possessives agree with the noun’s grammatical gender:

• min is used with en-words (common gender): min bok, min dagbok
• mitt is used with ett-words (neuter): mitt hus, mitt barn
• plural: mina: mina böcker

Since dagbok is an en-word (en dagbok), you use min.

Yes, Swedish adverbs like också can sometimes move depending on what you want to emphasize, but in the paired structure inte bara ..., utan också ..., the most standard placement is exactly as shown: right after utan.
If you move it, the sentence can sound less balanced or shift emphasis. For example, ... utan skriva också några rader ... is possible but usually feels less natural than utan också skriva ....

Yes, utan can mean: 1) without (preposition): utan socker = without sugar
2) but/rather (conjunction): inte X utan Y = not X but Y

Here it’s clearly the conjunction because it appears in the pattern inte bara ..., utan också ..., linking two alternatives/actions rather than modifying a noun like socker.