Learn Hardcore Chinese: wǒ xiǎng rènshi nǐ.

Here 想 (xiǎng) means would like/want to. It expresses desire. For “I think (that) …” use 我觉得 (wǒ juéde) or 我认为 (wǒ rènwéi). Note: 我想 can introduce an opinion in speech, but when followed by a verb like 认识, it means “want to.”

Is 我想认识你 natural? When would people say it?

It’s grammatical and clear, but quite direct. It’s most natural:

in dating contexts (sounds like “I’d like to get to know you”),
in networking/introductions online (still a bit forward). As a first line to a stranger, it can feel bold. People often soften it or phrase it as a question/request.

How can I make it more polite or less direct?

Commonly soften with permission words, “a bit,” or “we”:

我可以认识你吗？ / 我能认识你吗？ (May I…?)
我们可以认识一下吗？ (May we get to know each other a bit?)
我们认识一下吧。 (Let’s get to know each other.)
Add respect: 我想认识您。 (to elders/clients) Very natural in real life: 加个微信可以吗？ (Could we add each other on WeChat?)

What’s the difference between 认识, 知道, 了解, and 见/见面?

认识 (rènshi): to know someone/be acquainted; also to recognize characters/things. Example: 我认识他。 (I know him.)
知道 (zhīdào): to know a fact. 我知道他的名字。
了解 (liǎojiě): to know in depth/understand. 我了解他的性格。
见 (jiàn)/见面 (jiànmiàn): to see/meet face-to-face. 我想见你。 (I want to see you/meet you.) “Get to know” (become acquainted) is 认识, not 知道.

Should I add 一下 after 认识?

Adding 一下 (yíxià) softens the action: 认识一下 ≈ “get to know (a bit)”. Natural patterns:

我们认识一下吧。
可以跟你认识一下吗？ Avoid the clunky 认识你一下; say 认识一下你 sparingly, or better, use the versions above.

Can I say 我要认识你?

Grammatically yes, but 要 (yào) sounds strong/insistent (“I’m going to/I demand to”). It can feel pushy. 想 (xiǎng) is safer and more polite here.

How do I turn it into “May I/Can we get to know each other?”

我可以认识你吗？
我能认识你吗？
我们可以认识一下吗？
Even softer: 方便认识一下吗？ / 方便加个微信吗？

Should I use 你 or 您?

你 (nǐ): normal/informal “you”.
您 (nín): polite/respectful to elders, clients, teachers.
So 我想认识您 is respectful, but in a romantic context it may sound overly formal or distant.

How should I pronounce this? Any tone changes?

Pronunciation: wǒ xiǎng rènshi nǐ.
Tone sandhi: Two third tones in a row cause the first to rise, so 我想 is pronounced roughly wó xiǎng. 认识 is rènshi where shi is neutral tone. Pay attention to retroflex sounds: r in rèn and sh in shi.

What’s the grammar pattern here?

It’s the common pattern: Subject + modal + verb + object.

我 (subject) + 想 (modal “want to”) + 认识 (verb “get to know”) + 你 (object).
No extra particles are required.

Do I need 跟, like 跟你认识?

Base form is transitive: 认识你 is perfectly fine. You can add 跟 (gēn) when using 一下 to emphasize “with you”:

可以跟你认识一下吗？ (natural)
Plain 认识你 is already correct.

How do I say “Nice to meet you”?

Use:

很高兴认识你。 (Nice to meet you.)
认识你很高兴。
These are for after you’ve met/been introduced, not as an opener expressing desire.

How could someone respond to 我想认识你 politely?

Positive: 好的，我们认识一下吧。 / 行，加个微信吧。
Neutral: 可以，先自我介绍一下吧。
Polite decline: 不好意思，不太方便。 / 谢谢，但我就不打扰了。

Can I use 了 with 认识 to mean “got to know”?

Yes, for completed action:

我认识了一个新朋友。 (I got to know a new friend.)
我们昨天认识了。 (We got acquainted yesterday.) Don’t say 把你认识了—that’s ungrammatical.

Any common mistakes to avoid with this sentence?

Confusing 我想认识你 with 我想你; the latter means “I miss you.”
Using 知道 for people: 我知道你 usually means “I know of you,” not “I know you personally.”
Overusing 要: 我要认识你 can sound forceful.
Awkward phrasing 认识你一下; prefer 我们认识一下 or 跟你认识一下.

How do I refer to multiple people?

Use plural 你们 (nǐmen):

我想认识你们。 (I’d like to get to know you all.)
我们可以认识一下吗，各位？
Avoid 您们 in most contexts; it’s rarely used.

Any synonyms or alternative phrasing?

Formal/literary: 我想结识您。 (I’d like to make your acquaintance.)
Casual/social: 想跟你交个朋友。 (I’d like to be friends.)
Practical modern approach: 加个微信可以吗？ (Shall we add each other on WeChat?)