Learn Hardcore Japanese: watasi ha nihongo no benkyou wo akiramemasen.

So 私は doesn’t literally mean “I am,” but “as for me,” and then the rest of the sentence tells you something about that topic: that you won’t give up studying Japanese.
In everyday English we usually translate it simply as “I”, but the nuance is “regarding me…”

Could I use が instead of は and say 私が日本語の勉強をあきらめません?

You can say that, but the nuance changes.

私は日本語の勉強をあきらめません。
Neutral statement: “I will not give up studying Japanese.” (Topic: me)
私が日本語の勉強をあきらめません。
Emphasizes that it is I (and not someone else) who won’t give up.
This could be used when contrasting with others:
“Everyone else might give up, but I won’t.”

For a normal, neutral sentence, 私は is more natural.

Why do we say 日本語の勉強 instead of 日本語を勉強 here?

They are different structures:

日本語の勉強 = “study of Japanese” (a noun phrase: language + の + study)
日本語を勉強する = “to study Japanese” (a verb phrase: object + を + verb)

In this sentence, 勉強 is being treated as a noun, and it’s the thing that you “do not give up.”
So you get:

日本語の勉強 (the study of Japanese) + を (object marker) + あきらめません (do not give up).

If you changed it to a verb phrase, you’d say something like:

私は日本語を勉強することをあきらめません。
(“I will not give up studying Japanese” – literally, “I will not give up the act of studying Japanese.”)

What is the role of の in 日本語の勉強?

の here links two nouns:

日本語 = Japanese (language)
勉強 = study

日本語の勉強 literally means “study of Japanese.”
In English we often show this with “of” or by putting one noun in front of the other:

study of Japanese
Japanese study

So の is functioning like an “of” that shows what kind of study it is.

Why is there an を after 勉強? I thought を usually comes after things like 日本語.

を always marks the direct object of a verb, not specifically languages.

In this sentence, ask: “What is being ‘given up’?”

The thing being given up is 日本語の勉強 (the study of Japanese).

So the direct object of あきらめません is 日本語の勉強, and を attaches to the last noun in that noun phrase:

日本語　の　勉強　を　あきらめません
(I will not give up the study of Japanese.)

The whole phrase 日本語の勉強 acts as one object, and を attaches to 勉強 at the end.

What is the base (dictionary) form of あきらめません, and what kind of conjugation is this?

The dictionary form is あきらめる (to give up / abandon).
あきらめません is:

polite
present / future tense
negative

Conjugation steps (polite form):

Dictionary form: あきらめる
Polite present: あきらめます (“give up / will give up”)
Polite negative: あきらめません (“do not give up / will not give up”)

Does あきらめません mean present tense or future tense?

In Japanese, the same form is used for both simple present and future.

あきらめません can mean:

“I do not give up (in general).”
“I will not give up (from now / in this situation).”

Context decides whether you understand it as present habit or future intention.
In a sentence like this, it’s naturally interpreted as future intention:
“I’m not going to give up studying Japanese.”

Can I drop 私 and just say 日本語の勉強をあきらめません?

Yes, and that’s very natural in Japanese.

If the context already makes it clear that you are talking about yourself, you normally omit the subject:

日本語の勉強をあきらめません。
→ “(I) won’t give up studying Japanese.”

Japanese often leaves out 私 when it’s obvious from context. Using 私 is not wrong, but it can sound more explicit or slightly formal, like you’re emphasizing yourself as the topic.

Is it okay to say 日本語をあきらめません instead? What is the difference?

You can say 日本語をあきらめません, but the nuance shifts:

日本語の勉強をあきらめません。
→ “I won’t give up studying Japanese.”
Focus is on the study/learning process.
日本語をあきらめません。
→ Literally “I won’t give up Japanese.”
This is a bit broader or more emotional: not giving up on the language itself (your relationship to Japanese, learning, maybe using it, etc.).

For a clear, practical “I won’t give up studying Japanese,”
日本語の勉強をあきらめません is the most straightforward.

Why is 勉強 a noun here and not the verb 勉強します?

勉強 can function as both:

a noun: “study, studies”
part of a verb phrase when combined with する: 勉強する = “to study”

In 日本語の勉強をあきらめません, we need a thing that can be given up.
Verbs cannot take をあきらめる directly, so we use the noun:

(the) study of Japanese (日本語の勉強) + を + give up (あきらめる)

If you keep 勉強 as a verb, you must first turn it into a noun-like expression, for example:

日本語を勉強することをあきらめません。
(“I will not give up the act of studying Japanese.”)

How polite is あきらめません? Is this casual or formal?

あきらめません is the polite form, used with ます / ません.

Polite positive: あきらめます
Polite negative: あきらめません

For casual speech, you’d usually use the plain form:

Plain positive: あきらめる
Plain negative: あきらめない

So a casual version of the whole sentence would be:

日本語の勉強をあきらめない。
→ “I won’t give up studying Japanese.” (informal)

Is 私 always necessary in Japanese when I mean “I”?

No. 私 is not always necessary.

Japanese often omits subjects (and even objects) when they can be understood from context. Common patterns:

At the start of a conversation, or when you first introduce yourself, you might say:
私は日本語の勉強をあきらめません。
Later, once it’s clear you’re still talking about yourself, you would more naturally say simply:
日本語の勉強をあきらめません。

Using 私 makes the subject explicit, but not using it is often more natural and less repetitive in normal conversation.

Are there any more emphatic ways to say “I definitely will not give up” in Japanese based on this sentence?

Yes, you can add words for emphasis. Starting from:

私は日本語の勉強をあきらめません。

You could say, for example:

私は絶対に日本語の勉強をあきらめません。
→ “I will absolutely not give up studying Japanese.”
私は日本語の勉強をあきらめるつもりはありません。
→ “I have no intention of giving up studying Japanese.”

The core grammar stays the same; you just add adverbs or phrases like 絶対に (absolutely) or つもりはありません (have no intention).