Learn Hardcore Japanese: konsyuu ha nihongo no benkyouzikan wo huyasitai desu. - This week I want to increase my Japanese study time.

今週は marks “this week” as the topic of the sentence: as for this week….
You’re saying: “(As for) this week, I want to increase my Japanese study time.”

が would mark a grammatical subject, and with time expressions that’s usually not what you want here.
に is sometimes used with specific points in time (e.g. 誕生日に, 三時に), but with general periods like 今週, 来年, 昨日, Japanese normally just puts the time word in front with no particle, or uses は to make it the topic.

So:

今週日本語の勉強時間を増やしたいです。✔
今週は日本語の勉強時間を増やしたいです。✔ (adds “this week (especially)” nuance)
今週に日本語の勉強時間を増やしたいです。✖ unnatural in normal speech

Where is the word “I”? Why is 私 not written?

Japanese often omits the subject when it’s clear from context.

In this sentence, the person who wants to increase study time is clearly the speaker, so “I” is understood without saying 私. If you want to be very explicit, you can say:

私は今週は日本語の勉強時間を増やしたいです。

But having two は in a row (私は今週は…) can sound a bit heavy; usually people would say either:

今週は日本語の勉強時間を増やしたいです。 (topic = this week)
or
私は日本語の勉強時間を今週増やしたいです。 (topic = I)

Why is it 日本語の勉強時間 and not just 日本語勉強時間 or 勉強の時間?

日本語の勉強時間 literally means “the study time of Japanese”, i.e. “time spent studying Japanese.”

日本語の modifies 勉強時間 using の, which works like “of” or a possessive/linking marker.
日本語勉強時間 without の is not natural; Japanese doesn’t usually just stack that many nouns without の in the middle.
勉強の時間 is fine grammatically, but it just means “study time” in general. If you want to be clear that it’s Japanese study time, 日本語の勉強時間 is better.

You could also say:

日本語を勉強する時間 = “the time (that I) study Japanese”
(a relative clause instead of a compound noun)

Why is を used after 勉強時間, not が?

Here, 勉強時間 is the direct object of the verb meaning “increase.”

With a transitive verb (a verb that acts on something), the “something” is marked by を.
So you are increasing (what?) → 勉強時間, so you use を.

If the verb were intransitive (“to increase” by itself), 勉強時間 would be the subject and take が:

勉強時間が増える。= “The study time increases.” (by itself, not “I increase it”)
勉強時間を増やす。= “(I) increase the study time.”

What is the difference between focusing on 勉強時間を増したい and saying something like もっと日本語を勉強したい?

Both talk about “doing more Japanese study,” but the focus is different:

勉強時間を増したい / 増やしたい focuses on the amount of time.
→ “I want to increase the number of hours I study Japanese.”
もっと日本語を勉強したい focuses more on the overall quantity/intensity of study.
→ “I want to study Japanese more (more often / more seriously / more in general).”

They often overlap in meaning, but if you specifically mean more hours, 勉強時間を増やしたい is clearer.

How is 増したい formed? Is it related to the past tense 増した?

The verb is 増す (“to increase”). To make the -たい form (“want to do”), you attach たい to the ます-stem of the verb.

Dictionary form: 増す
ます-stem (the part before ます): 増し (from 増します)
- たい → 増したい

So 増したい = “want to increase.”
It looks like past tense (増した), but grammatically it’s different:

増した = past (increased)
増したい = want to increase

The “した” here comes from 増し + たい, not from する or past tense.

What’s the difference between 増す, 増やす, and 増える? Which sounds most natural here?

増える (ふえる) = intransitive: “to increase (by itself)”
- 勉強時間が増える。= “Study time increases.”
増やす (ふやす) = transitive: “to increase something”
- 勉強時間を増やす。= “(I) increase my study time.”
増す (ます) can be both, but it’s often more formal or literary, and used a lot with abstract things (興味が増す, 不安が増す).

In everyday conversation, for this meaning, 今週は日本語の勉強時間を増やしたいです。 is more natural and common than using 増す. Many native speakers would instinctively say 増やしたい here.

What exactly does たいです mean? Is it a verb or an adjective? How polite is it?

～たい is grammatically an い-adjective that attaches to a verb stem and expresses the speaker’s desire:

増したい = “want to increase”
食べたい = “want to eat”
行きたい = “want to go”

It conjugates like an い-adjective:

増したい, 増したくない, 増したかった, etc.

Adding です makes it polite:

増したい。= plain / casual
増したいです。= polite

So 増したいです is “(I) want to increase (it)” in polite speech.

Can I use たいです to talk about someone else’s desires?

In principle, たい expresses the speaker’s own desire. Using it directly for someone else can sound like you are speaking for them or guessing their inner feelings, which is considered a bit too direct.

Instead, for third person you usually use たがる / たがっています:

田中さんは日本語の勉強時間を増やしたがっています。
= “Tanaka seems to want to increase his/her Japanese study time.”

You can hear things like:

田中さんも勉強時間を増やしたいと言っています。
= “Tanaka also says he/she wants to increase study time.”

Here you quote what the person themselves says, which is fine.

Can I omit です or use だ instead?

Yes, but it changes the level of politeness.

今週は日本語の勉強時間を増やしたい。
→ Plain form, casual; good with friends, diaries, etc.
今週は日本語の勉強時間を増やしたいです。
→ Polite; good in most normal situations (with teachers, coworkers, etc.).
今週は日本語の勉強時間を増やしたいだ。
→ Incorrect. たい is an い-adjective, so it does not take だ directly.

If you need a plain copula with たい (for grammar reasons), you’d treat たい as an い-adjective:

増やしたいんだ / 増やしたいのだ (explanatory “it’s that I want to increase …”)

Can I change the word order, like 日本語の勉強時間を今週は増したいです?

Yes, Japanese word order is relatively flexible as long as the particles stay attached to the right words.

All of these are grammatically fine (with 増やしたい for naturalness):

今週は日本語の勉強時間を増やしたいです。
日本語の勉強時間を今週は増やしたいです。
日本語の勉強時間を増やしたいです、今週は。 (last one is more conversational/emphatic)

The default and most neutral is usually to put the topic (は) near the beginning, as in the original sentence.

How can this sentence talk about the future if there is no future tense in Japanese?

Japanese generally uses the non-past form (dictionary form / ます-form) for both present and future.

Here, 増したいです / 増やしたいです is non-past, so the time expression 今週 tells you it’s about the near future:

今週は日本語の勉強時間を増やしたいです。
= “This week, I want to increase my Japanese study time.” (a plan/hope for this week)

So the idea of “future” comes from context and time words like 今週, 明日, 来年, not from a special verb tense.