Janeiro é o mês em que eu começo a organizar minhas férias.

Yes. In Portuguese, names of months are normally written in lowercase: janeiro, fevereiro, março, etc.

Here it is capitalized only because it is the first word of the sentence.
So inside a sentence, you would normally write:

Eu viajo em janeiro.

not

Eu viajo em Janeiro.

In Portuguese, months usually appear without an article when you are simply naming them:

Janeiro é o mês... = January is the month...

That is normal and natural.

In some contexts, you may see an article with months, especially in more specific expressions or in some regional/stylistic uses, but here Janeiro without o is the standard choice.

Because the sentence is identifying January as a specific month:

Janeiro é o mês em que... = January is the month when...

Using o mês makes it definite: the month.
If you said um mês, it would mean a month, which does not fit the idea here.

Em que here means in which or more naturally in English, when.

So:

o mês em que eu começo...
literally = the month in which I start...
natural English = the month when I start...

This structure is very common in Portuguese after words for time or place:

• o dia em que = the day when/on which
• o ano em que = the year when/in which
• a cidade em que = the city where/in which

In standard Portuguese, o mês em que eu começo is the safest and most grammatical form.

People do sometimes omit the preposition in informal speech, but after mês, the more correct and expected structure is em que.

You could also hear:

Janeiro é o mês quando eu começo...

but quando is generally less preferred here than em que in careful standard Portuguese.

So for learners, the best model is:

o mês em que...

Yes, it could. Portuguese often drops subject pronouns when the verb ending already makes the subject clear.

So both are possible:

• eu começo
• começo

Since começo clearly means I start, the eu is optional.

Why include eu then?

Common reasons:

• emphasis
• clarity
• rhythm/style
• contrast with someone else

In this sentence, eu is not required, but it sounds perfectly natural.

After começar, Portuguese often uses a + infinitive to mean to start doing something.

So:

• começar a estudar = to start studying
• começar a trabalhar = to start working
• começar a organizar = to start organizing

This is a very common pattern.

You may also see começar por + infinitive in some contexts, but here começar a organizar is the normal choice.

Começo is the 1st person singular present indicative of começar:

• eu começo = I start
• você começa = you start
• ele/ela começa = he/she starts
• nós começamos = we start
• eles/elas começam = they start

Notice the spelling change: the infinitive is começar, but with eu, it becomes começo with ç.
That spelling keeps the s sound before o.

Minhas agrees with férias.

• férias is feminine plural
• so the possessive must also be feminine plural

That is why we get:

• meu = masculine singular
• minha = feminine singular
• meus = masculine plural
• minhas = feminine plural

Since férias is feminine plural, the correct form is minhas férias.

Because in Portuguese, férias is normally a plural noun when it means vacation / time off from work or school.

So Portuguese says:

• minhas férias
• suas férias
• as férias de julho

Even if English uses singular vacation, Portuguese still uses férias in the plural.

This is one of those vocabulary differences you simply have to get used to.

Literally, it is organize my vacation or organize my vacations/time off.

But in natural usage, organizar minhas férias often means something like:

• plan my vacation
• start arranging my vacation
• get my vacation plans in order

It can include things like booking tickets, choosing dates, making reservations, and planning activities.

So the exact English wording depends on context, but the Portuguese expression is very natural.

Yes, completely normal.

The sentence structure is:

Janeiro + é + o mês + em que + eu começo + a organizar + minhas férias

A rough breakdown:

• Janeiro = subject
• é = is
• o mês = the month
• em que = in which / when
• eu começo = I start
• a organizar minhas férias = to organize my vacation / vacation plans

This is a very natural and standard way to express the idea in Portuguese.

Yes. The original sentence is already natural, but in conversation, people might also say things like:

• Em janeiro, eu começo a organizar minhas férias.
• Janeiro é quando eu começo a organizar minhas férias.
• Eu começo a organizar minhas férias em janeiro.

All of these are natural, but they shift the focus a little:

• Janeiro é o mês em que... emphasizes January as the defining month
• Em janeiro... emphasizes the time
• Eu começo... em janeiro emphasizes the action first

So the original sentence is correct and natural, just a bit more structured in tone.