Learn Hardcore Japanese: raigetu no ryokou no tame ni ookina suutukeesu wo kaimasita. - I bought a big suitcase for next month’s trip.

Japanese often omits the subject when it is obvious from context.
In this sentence, the speaker is clearly talking about their own action, so 私 (I) is left out.
If you really want to include it, you can say:

私は来月の旅行のために大きなスーツケースを買いました。

But even that 私は would usually be omitted in natural conversation unless you need to contrast yourself with someone else.

What does the particle の do between 来月 and 旅行?

の links two nouns in a way similar to the English ’s or of, but more general.
So:

来月の旅行 = the trip of next month → next month’s trip

Here 来月 (next month) modifies 旅行 (trip) via の, turning the whole thing into one noun phrase: 来月の旅行 (next month’s trip).

Why are there two の in 来月の旅行のために?

You actually have a chain of noun connections:

来月の旅行 = next month’s trip
旅行のため = for the sake of the trip

Put together:

来月の旅行 (next month’s trip) + のために (for the sake of / for)
→ 来月の旅行のために = for next month’s trip

Each の connects one noun to the next element. This kind of chaining is very common in Japanese.

What exactly does ために mean here?

ために (ために) literally means for the sake of or for the purpose of.

Common patterns:

Noun + のために → for (the sake/purpose of) that noun
- 旅行のために = for the trip
Verb (dictionary form) + ために → in order to (do that verb)
- 旅行するために = in order to travel

In this sentence, 来月の旅行のために means for next month’s trip, i.e., that trip is the purpose/reason for buying the suitcase.

Why do we need に after ため? Could we just say ため?

Here, ため is a noun meaning sake / purpose / benefit / reason, and に is the particle that attaches to it.
Together, ために acts almost like one set phrase meaning for / in order to / because of.

X のために Y する = do Y for/because of X

You will see ため by itself in some fixed expressions or more formal written styles, but in a normal sentence like this, you say ために, not just ため.

What is the difference between 大きな and 大きい? Could I say 大きいスーツケース?

Yes, you can say 大きいスーツケース. Both are correct.

大きい is the regular い-adjective form.
- 大きいスーツケース = a big suitcase
大きな is a special pre-noun form used only before nouns.
- 大きなスーツケース = a big suitcase (slightly more literary/story-like feel)

In everyday speech, 大きいスーツケース is very common. 大きなスーツケース sounds a bit more written, descriptive, or formal, but the difference in meaning is basically nil.

Why is スーツケース written in katakana?

Katakana is used mainly for:

Loanwords from other languages
Onomatopoeia
Some emphasis or stylistic effects

スーツケース comes from the English word suitcase, so it is written in katakana.
Most everyday foreign objects like テーブル (table), コンピューター (computer) are also written in katakana.

What does the particle を do before 買いました?

を marks the direct object of the verb – the thing that the action is done to.

スーツケースを買いました。
- スーツケース = suitcase (object)
- を = object marker
- 買いました = bought

So スーツケースを買いました means (I) bought a suitcase.
In the full sentence, the long phrase 来月の旅行のために gives the reason/purpose, and スーツケースを is the thing actually bought.

Why is it 買いました (past) and not 買います (non-past)?

買いました is the polite past tense of 買う (to buy):

買います = (I) buy / will buy
買いました = (I) bought / have bought

Using 買いました means the buying is already completed.
So the sentence means I bought a large suitcase for next month’s trip (it’s already done).

If you want to say I will buy a large suitcase for next month’s trip, you would use:

来月の旅行のために大きなスーツケースを買います。

Can I change the word order, like putting 来月の旅行のために later in the sentence?

Yes, as long as the verb stays at the end, Japanese word order is fairly flexible. For example:

来月の旅行のために大きなスーツケースを買いました。 (original)
大きなスーツケースを来月の旅行のために買いました。 (also OK)

The second version puts a bit more focus on what you bought first (大きなスーツケース), then adds the purpose.
Both are natural; the original is probably the most neutral ordering.

Can I say 来月旅行のために without the first の?

Not in this sentence; it would sound unnatural.

来月の旅行 = next month’s trip (one noun phrase)
来月旅行 is not a normal noun phrase meaning next month’s trip.

You can say things like:

来月旅行に行きます。 = Next month, (I) will go on a trip.

But there 来月 is modifying the verb phrase 旅行に行きます, not directly modifying 旅行 as a noun.

To say next month’s trip as a noun, you normally need 来月の旅行 with の.

Why is there no particle directly after 来月?

来月 is not standing alone here; it’s part of the noun phrase 来月の旅行.

The structure is:

来月
- の
  - 旅行
    - 来月 modifies 旅行 through の
    - so there’s no separate particle after 来月

If 来月 were acting as a time expression for the verb directly, you might see something like 来月、旅行に行きます。, where 来月 can even appear with a comma and no particle. But in 来月の旅行, the の already does the linking job.

What’s the difference between 旅行のために and something like 旅行に行く?

They express different relationships with 旅行 (trip/travel):

旅行のために
- のために = for the sake/purpose of
- 旅行のために = for the trip / for traveling
- Used to show purpose or reason for another action
  - 旅行のためにスーツケースを買いました。
    = I bought a suitcase for the trip.
旅行に行く
- に here marks destination/goal
- 旅行に行く = go on a trip

So 旅行のために explains why you do something, while 旅行に in 旅行に行く shows where/for what activity you are going.