Learn Hardcore Spanish (Spain): Quiero escribir la nota en mi cuaderno.

Quiero already means I want (first person singular).
Adding yo is not wrong, but it’s normally only used for emphasis or contrast:
- Yo quiero escribir la nota, no tú. – I want to write the note, not you.

So:

Quiero escribir la nota… = natural, neutral.
Yo quiero escribir la nota… = “I (specifically) want to write the note…” (emphasising the subject).

Why do we use quiero + escribir (an infinitive)? Could I say something like quiero que escribo?

With querer followed by another verb, Spanish normally uses the infinitive:

Quiero escribir – I want to write
(literally: I-want to-write)

Using quiero que escribo is incorrect. If you use que, the structure changes:

Quiero escribir la nota. – I want to write the note (myself).
Quiero que escribas la nota. – I want you to write the note.

Notice:

After quiero que, the verb escribir is conjugated in the subjunctive (escribas), and it usually refers to someone else doing the action.

Why is it la nota and not una nota?

Both are possible; they just give slightly different meanings:

La nota = the note (a specific note that both speaker and listener already know about or can identify from context).
- Quiero escribir *la nota en mi cuaderno.
  → I want to write **the note in my notebook* (maybe a note the teacher just dictated).
Una nota = a note (non-specific / not yet identified).
- Quiero escribir *una nota en mi cuaderno.
  → I want to write **a note in my notebook* (any note; you’re introducing it).

So the choice between la and una depends on whether the note is already known/specific (la) or being mentioned for the first time/unspecific (una).

Is nota always “note” like a written note, or can it mean other things?

Nota in Spanish is quite flexible. It can mean:

Written note
- Escribí una nota para recordarlo. – I wrote a note to remember it.
Grade/mark in school
- He sacado una buena nota en el examen. – I got a good grade on the exam.
Musical note
- No puedo cantar esa nota. – I can’t sing that note.
Short news item / announcement
- Leí una nota en el periódico. – I read a short piece in the newspaper.

In your sentence, la nota is naturally understood as a written note or a brief written message.

Why is it mi cuaderno and not mío cuaderno?

Spanish has two types of possessive forms:

Short (unstressed) possessives, which go before the noun:
- mi cuaderno – my notebook
- tu cuaderno – your notebook
- su cuaderno – his/her/your (formal)/their notebook
- nuestro cuaderno – our notebook
Long (stressed) possessives, which usually go after the noun (or stand alone):
- el cuaderno mío – the notebook of mine / my notebook (emphatic)
- el mío – mine (standing alone, without a noun after it)

So:

mi cuaderno = correct, normal.
mío cuaderno = incorrect.
el cuaderno mío = correct but more emphatic or contrastive.

Why is it mi cuaderno and not mis cuaderno?

The possessive mi/mis agrees with the number of the noun it modifies:

mi
- singular noun:
  - mi cuaderno – my notebook
mis
- plural noun:
  - mis cuadernos – my notebooks

Since cuaderno is singular, you must use mi, not mis.

How do I know that cuaderno is masculine and needs mi cuaderno, not something like mia cuaderno?

In Spanish, every noun has a grammatical gender (masculine or feminine), which you usually have to memorize. However, there are patterns:

Nouns ending in -o are usually masculine.
- el cuaderno – the notebook
- el libro – the book
- el vaso – the glass

Because cuaderno is masculine:

The definite article is el: el cuaderno.
The possessive form is mi cuaderno (the possessive doesn’t change form for gender in the singular; mi is used for both masculine and feminine singular nouns).

So:

mi cuaderno – my notebook (masculine, singular)
mi mesa – my table (feminine, singular)
mis cuadernos / mis mesas – my notebooks / my tables (plural for both).

Could I say en el cuaderno instead of en mi cuaderno?

Yes, grammatically you can say both; they just have different meanings:

en mi cuaderno – in my notebook (the notebook belongs to the speaker).
en el cuaderno – in the notebook (a specific notebook, but ownership is not specified; it might be shared, the teacher’s notebook, a notebook just mentioned, etc.).

Example contrast:

Quiero escribir la nota en *mi cuaderno. – I want to write the note in *my notebook.
Quiero escribir la nota en *el cuaderno del profesor. – I want to write the note in *the teacher’s notebook.

Why do we say en mi cuaderno and not en mi cuaderno a or something like that? What does en cover exactly?

In this sentence, en covers the meanings of:

in: inside something
on: on the surface of something

Spanish often uses en where English distinguishes between in and on:

en la mesa – on the table
en la caja – in the box
en mi cuaderno – in/on my notebook (written on its pages)

You do not add an extra preposition like a here. The correct structure is:

escribir algo en algo – to write something in/on something
e.g. Escribo la dirección en el papel. – I write the address on the paper.

Is the word order fixed? Could I say En mi cuaderno quiero escribir la nota?

The basic, most neutral order is:

Quiero escribir la nota en mi cuaderno.

But Spanish word order is relatively flexible. You can move parts for emphasis or style:

En mi cuaderno quiero escribir la nota.
Sounds like you’re emphasising in my notebook (maybe because someone suggested another place).

Other possible (but less neutral) orders:

La nota quiero escribirla en mi cuaderno. – emphasises the note.
La quiero escribir en mi cuaderno, la nota. – more conversational/emphatic.

All are grammatically possible, but your original sentence is the most straightforward.

Can escribir be conjugated here? Why not quiero escribo la nota?

After querer (when it expresses wanting to do an action yourself), Spanish uses the infinitive, not a conjugated verb:

Quiero escribir la nota. – I want to write the note.

Using quiero escribo is incorrect, because that would be two finite (conjugated) verbs in a row, which Spanish doesn’t allow in this structure.

Compare:

Quiero escribir. – I want to write.
Quiero que escribas. – I want (that) you write.
(Here quiero is conjugated and escribas is conjugated in the subjunctive, but they are in two different clauses: quiero / que escribas.)

How do you conjugate querer in the present tense, and is its irregularity visible in quiero?

Yes, quiero shows the irregularity of querer. Present indicative of querer:

yo quiero – I want
tú quieres – you want (informal singular)
él / ella / usted quiere – he / she wants; you want (formal singular)
nosotros / nosotras queremos – we want
vosotros / vosotras queréis – you all want (informal plural, mainly Spain)
ellos / ellas / ustedes quieren – they want; you all want (formal plural, and also standard plural in Latin America)

The stem changes from quer- to quier- in all forms except nosotros and vosotros.

In Spain, is cuaderno the usual word for “notebook,” or are there other common options?

In Spain, cuaderno is very common and standard for notebook (especially school notebooks). Other words you may also hear:

libreta – also means notebook; quite common and can sound a bit more informal or regional depending on area.
bloc (or bloc de notas) – more like a notepad/pad of paper, often with a spiral.

In your sentence:

Quiero escribir la nota en mi cuaderno. – perfectly natural in Spain.
Quiero escribir la nota en mi libreta. – also natural; just a slightly different preference of word.