Learn Hardcore Spanish (Spain): Quiero estudiar tranquilo en mi cuarto.

Quiero already tells you it’s “I want” (1st person singular), so yo is not necessary.
Yo quiero estudiar tranquilo en mi cuarto is also correct, but it sounds more emphatic, like “I want to study…” (maybe in contrast to someone else).

So the most natural everyday version is exactly what you see: Quiero estudiar tranquilo en mi cuarto.

Why is it tranquilo and not tranquilamente? Shouldn’t an adverb modify the verb estudiar?

Both are possible, but they’re not quite the same:

Quiero estudiar tranquilo en mi cuarto.
Literally “I want to study calm / relaxed in my room.”
Here tranquilo describes your state while studying. It’s like saying “I want to study while I’m feeling calm / at peace.”
Quiero estudiar tranquilamente en mi cuarto.
“I want to study calmly / in a calm way in my room.”
Here tranquilamente is a standard adverb emphasizing the manner of studying.

In everyday Spanish (especially in Spain), using an adjective like tranquilo with verbs such as trabajar, estudiar, dormir is very common to describe how the subject feels while doing the action:

Quiero dormir tranquilo. – I want to sleep peacefully.
Prefiero trabajar tranquilo. – I prefer to work in peace.

So tranquilo here is completely natural and very idiomatic.

Does tranquilo agree with yo or with cuarto?

It agrees with the subject, which is (yo), not with cuarto.

Subject: (yo) → masculine singular (or default/unspecified)
Adjective: tranquilo → masculine singular

Even if the speaker is a woman, it’s still common and correct for her to say:

Quiero estudiar tranquila en mi cuarto. (feminine speaker, matching yo)
Quiero estudiar tranquilo en mi cuarto. (masculine speaker, matching yo)

Cuarto is also masculine singular, but in this sentence tranquilo is not describing the room; it’s describing how the person will be / feel while studying.

If you wanted to describe the room, you’d need a different structure, for example:

Quiero estudiar en un cuarto tranquilo. – I want to study in a calm/quiet room.

If I’m a woman, do I have to say tranquila?

No one will be confused if you say tranquilo, but the grammatically “expected” agreement is:

Man speaking: Quiero estudiar tranquilo en mi cuarto.
Woman speaking: Quiero estudiar tranquila en mi cuarto.

Native speakers, especially in quick speech, sometimes don’t mark gender consistently, but as a learner it’s good practice to match:

Yo (feminine) → tranquila
Yo (masculine) → tranquilo

Could I say Quiero estar tranquilo en mi cuarto instead? Is that different?

Yes, and the nuance changes slightly:

Quiero estudiar tranquilo en mi cuarto.
Focus on the activity: you want to study, and to do so in a calm/peaceful state.
Quiero estar tranquilo en mi cuarto.
Focus on the state: you want to be calm in your room (maybe not necessarily studying, just being there peacefully).

In many real-life situations, both might fit, but:

If you’re emphasizing that you have to study: use estudiar.
If you’re emphasizing you just want peace and quiet: use estar tranquilo.

Why is it en mi cuarto and not a mi cuarto or something else?

Because en is the normal preposition for location in Spanish:

en mi cuarto – in my room
en casa – at home
en el coche – in the car

a is usually used for direction / movement:

Voy a mi cuarto. – I’m going to my room.
Subo a mi habitación. – I’m going up to my room.

In your sentence there is no movement; you’re just saying where you want to study, so en mi cuarto is the correct choice.

Does mi cuarto mean “my bedroom” or just “my room” in general? And what about habitación?

In Spain:

mi cuarto very often means “my bedroom”, especially in everyday family contexts (parents talking to children, roommates, etc.).
mi habitación also means “my (bed)room” and is maybe a bit more neutral or “standard.”

Many speakers in Spain use cuarto and habitación almost interchangeably when talking about a bedroom:

Estoy en mi cuarto. – I’m in my room/bedroom.
Voy a mi habitación. – I’m going to my room/bedroom.

If you literally mean just “a room” (not specifically a bedroom), context usually makes it clear, or you might say:

una sala – a living room / lounge
un despacho – an office / study
una habitación – a room (in a hotel, in a house, etc.)

Why is it quiero and not a future form like querré if I’m talking about what I want to do?

Because you’re expressing a present desire, not a future prediction.

Quiero estudiar tranquilo en mi cuarto.
= Right now, this is what I want.

The future querré (“I will want”) is used much less often and usually to predict a future desire:

Mañana querré descansar. – Tomorrow I’ll (probably) want to rest.

In practice, to talk about what you wish/intend to do (now or in the near future), native speakers use quiero + infinitive:

Quiero estudiar. – I want to study.
Quiero salir esta noche. – I want to go out tonight.

Is Quiero estudiar en mi cuarto tranquilo also correct? Where should tranquilo go?

Quiero estudiar en mi cuarto tranquilo is grammatically possible, but it changes the meaning:

Quiero estudiar tranquilo en mi cuarto.
→ tranquilo describes you (the subject) while studying.
Quiero estudiar en mi cuarto tranquilo.
→ tranquilo now naturally describes cuarto (“my quiet/calm room”).

So:

If you want to highlight your peaceful state, put tranquilo right after estudiar.
If you want to highlight that the room is quiet, put tranquilo after cuarto or say:
- Quiero estudiar en un cuarto tranquilo.
- Quiero estudiar en mi cuarto, que es muy tranquilo.

Could I say Me gustaría estudiar tranquilo en mi cuarto? What’s the difference with Quiero?

Yes, you can, and the difference is mainly politeness / softness:

Quiero estudiar tranquilo en mi cuarto.
→ Direct: “I want to study in peace in my room.”
Sounds firm, neutral, quite common in speech.
Me gustaría estudiar tranquilo en mi cuarto.
→ Softer and more polite: “I would like to study in peace in my room.”
Sounds a bit more tentative or respectful, like a polite request.

In everyday Spanish, both are very common. Use me gustaría when you want to sound especially polite or less demanding.