Learn Hardcore Japanese: watasi ha rainen kaigairyokou wo suru tumori desu.

If you used が instead (私が), it would sound like you’re emphasizing who is going to travel, often in contrast to someone else, e.g. “I’m the one who is going to travel next year (not someone else).”
In normal, neutral self-introducing statements, 私は is standard, and in casual speech you can just drop 私 entirely.

Why doesn’t 来年 have a particle like に?

Time expressions like 来年 (next year), 明日 (tomorrow), 昨日 (yesterday), 毎日 (every day) can often be used without a particle when they function as adverbs of time:

来年海外旅行をするつもりです。
I intend to travel abroad next year.

A particle に is possible:

来年に海外旅行をするつもりです。

…but with broad time words like 来年, に is often omitted in everyday speech.
Using に sounds a bit more formal/fixed-schedule-like, and is more common with specific points in time:

３月に日本へ行きます。 – I will go to Japan in March.
来週の土曜日に行きます。 – I’ll go next Saturday.

Why does 海外旅行 take the object marker を even though it’s a noun?

In Japanese, many noun + する combinations act like a verb phrase:

勉強をする – to study
運動をする – to exercise
仕事をする – to work
海外旅行をする – to travel abroad

Here 海外旅行 is a noun, and する is “to do.”
So literally, 海外旅行をする is “to do overseas travel.”

Because 海外旅行 is the “thing you do,” it is marked with を as the direct object of する. Together, 海外旅行をする works almost like a single verb meaning “to travel abroad.”

Why is it 海外旅行をする instead of something like 海外に行く?

Both are possible, but they have slightly different nuances:

海外旅行をする – to take a trip abroad / to go on an overseas trip
- Focuses on the trip as an activity or event.
海外に行く – to go abroad
- Focuses on the act of going to a foreign country.

Your sentence:

来年海外旅行をするつもりです。
Suggests “I’m planning an overseas trip next year,” maybe for leisure/holiday.

If you say:

来年海外に行くつもりです。
It’s more neutral “I intend to go abroad next year,” without specifically emphasizing “a trip” feeling, though in context it may still be understood that way.

What exactly does つもりです mean, and how is it different from 予定です?

つもり expresses personal intention — what you (or someone) intend to do:

～するつもりです。 – I intend to do ~ / I’m planning to do ~ (in my mind).

予定 (よてい) expresses a plan/schedule that exists more objectively, like something on a calendar:

来年海外旅行をする予定です。
– There is a plan/schedule for me to travel abroad next year.

Nuance:

つもり = inner decision, will, intention.
予定 = arrangement, schedule (like booking tickets, arrangements with others, etc.).

So つもりです is often more about your personal wish/decision; 予定です suggests it’s been organized or is on the schedule.

Why is it するつもり and not しますつもり?

つもり is a noun. The part before it is a clause in plain/dictionary form that modifies this noun:

（海外旅行をする）つもり
literally: “the intention of doing overseas travel”

Before a noun like つもり, you normally use the plain form of the verb:

行くつもり (not 行きますつもり)
勉強するつもり (not 勉強しますつもり)

Politeness is expressed at the end of the sentence with です / ます, not inside the modifying clause:

Plain: 海外旅行をするつもりだ。
Polite: 海外旅行をするつもりです。

So しますつもりです is ungrammatical; keep する in plain form before つもり, and use です to be polite.

Why is a present form (する) used to talk about the future?

Japanese doesn’t have a distinct grammatical “future tense” like English. The non‑past form (dictionary form or ます-form) covers both present and future, and context tells you which is meant.

来年海外旅行をするつもりです。
→ Because of 来年 (next year) and つもり (intention), it clearly refers to the future.

Similarly:

明日映画を見ます。 – I will watch a movie tomorrow.
毎日映画を見ます。 – I (usually) watch movies every day.

Same verb form, but the time expression changes the meaning.

Can I drop 私 from this sentence?

Yes, and that’s very common and natural in Japanese:

来年海外旅行をするつもりです。

Since you are talking about your own intentions, the subject “I” is usually clear from context. Japanese often omits pronouns (私, あなた, 彼, etc.) when they’re obvious.

You might keep 私は when:

You want to contrast yourself with others:
私は来年海外旅行をするつもりです。
(As for me, I intend to travel abroad next year — maybe others don’t.)
You’re introducing yourself or clarifying who you are in a new context.

In casual speech, it’s very natural to leave it out.

How would I say “I don’t intend to travel abroad next year” or “I wasn’t planning to”?

Use the negative of つもり (or the verb before it) and/or change the tense:

Negative intention (present/future):
- 来年海外旅行をするつもりはありません。
  → I have no intention of traveling abroad next year. (polite, formal-ish)
- 来年海外旅行をするつもりはないです。
  → Same meaning, slightly more conversational.
Negative of the verb before つもり:
- 来年海外旅行をしないつもりです。
  → I intend not to travel abroad next year.
Past intention (“I was planning to, but…”):
- 来年海外旅行をするつもりでした。
  → I was planning to travel abroad next year.
- 来年海外旅行をするつもりはありませんでした。
  → I wasn’t intending to travel abroad next year.

You then often add a follow-up clause explaining what happened:
でも、予定が変わりました。 – But the plan changed.

What’s the difference between 来年海外旅行をするつもりです and 来年は海外旅行をするつもりです?

Adding は after 来年 makes 来年 a topic and gives it a slight contrast / emphasis:

来年海外旅行をするつもりです。
→ Neutral: I intend to travel abroad next year.
来年は海外旅行をするつもりです。
→ “As for next year, I intend to travel abroad.”
Implies a contrast like:
- This year I won’t / didn’t, but next year I will.
- Or compared to some other time.

So 来年は often implies “at least next year” or “specifically next year (not some other time).”

What politeness level is this sentence, and how would it look in casual speech?

Your sentence:

私は来年海外旅行をするつもりです。

is in polite form because of です.

Casual (plain) version:

来年海外旅行をするつもりだ。
Or with 私 in casual style: 俺は来年海外旅行をするつもりだ。 (male, informal) / 私は is still okay but can sound a bit formal even in plain style.

Note:

You do not change する to します before つもり (that would be wrong).
You change only the final copula:
- Polite: …つもりです。
- Casual: …つもりだ。 / …つもり。 (dropping だ is possible in some casual contexts).

Can I move the words around, like 私は海外旅行を来年するつもりです?

Japanese word order is somewhat flexible, but there are preferences.

Your original:

私は来年海外旅行をするつもりです。

is the most natural ordering:
[topic] + [time] + [object] + [verb + つもり + です]

私は海外旅行を来年するつもりです。 is grammatically possible but sounds marked and a bit unnatural in most contexts. Putting 来年 after 海外旅行を tends to put contrastive emphasis on the timing:

Like: “I’ll do the overseas trip next year (not this year).”

In everyday neutral speech, time expressions like 来年, 明日, etc. usually come before the object and close to the beginning of the sentence, so:

来年海外旅行をするつもりです。
is the best default word order.