Vorrei conoscere meglio il tuo vicino.

Because Italian usually drops subject pronouns when they are not needed.

• Vorrei already tells you the subject is I, because the verb ending shows first person singular.
• So Vorrei conoscere meglio il tuo vicino is perfectly natural.
• You could say Io vorrei conoscere meglio il tuo vicino, but that usually adds emphasis, like I would like to know your neighbor better.

Vorrei is the conditional form of volere (to want).

• voglio = I want
• vorrei = I would like

In this sentence, vorrei sounds softer, more polite, and less direct than voglio.

So:

• Voglio conoscere meglio il tuo vicino = I want to get to know your neighbor better
• Vorrei conoscere meglio il tuo vicino = I’d like to get to know your neighbor better

A native English speaker often wonders this because English also switches between I want and I’d like.

In Italian, vorrei is very common when you want to sound:

• polite
• tentative
• less demanding
• more natural in conversation

So vorrei is often the better choice when expressing a wish or desire in a gentle way.

Because in Italian, after volere and its forms, you normally use the infinitive directly.

So:

• vorrei conoscere = I would like to know / get to know
• literally: I would-like know

Italian does not need a separate word equivalent to English to in this structure.

Other examples:

• Vorrei mangiare = I would like to eat
• Vorrei andare = I would like to go

This is a very important distinction in Italian.

• conoscere = to know people, places, or things through familiarity; also to meet / get to know
• sapere = to know facts, information, or how to do something

So here:

• conoscere il tuo vicino = to know / get to know your neighbor
• sapere qualcosa sul tuo vicino = to know something about your neighbor

You use conoscere because the sentence is about becoming more familiar with a person, not about learning facts.

Meglio means better.

In this sentence, it modifies conoscere, so the idea is:

• conoscere meglio = to know better / to get to know better

This is a very common Italian way to express the idea of getting to know someone more deeply or more fully.

So Vorrei conoscere meglio il tuo vicino is very natural for I’d like to get to know your neighbor better.

Because in Italian, possessives usually go together with a definite article.

So normally you say:

• il mio amico
• la tua macchina
• il tuo vicino

Italian often works like:

• the your neighbor
  rather than simply your neighbor

There is a well-known exception with many singular close family members, such as:

• mia madre
• tuo fratello

But vicino is not a family member, so the article il is needed.

Yes, vicino can be either:

• a noun: neighbor
• an adjective: near / close

In this sentence, it is a noun, meaning neighbor.

You can tell from the structure:

• il tuo vicino = your neighbor

If it were an adjective, it would be part of a different construction, such as:

• La scuola è vicina = The school is nearby
• Abito vicino = I live nearby

Yes. In this sentence, vicino is masculine singular, so it refers to a male neighbor.

If you mean a female neighbor, you would say:

• Vorrei conoscere meglio la tua vicina

So the ending changes:

• vicino = male neighbor
• vicina = female neighbor

Tuo is the possessive form used with tu, so it is informal singular: your when speaking to one person informally.

If you were speaking formally to one person, you would usually say:

• Vorrei conoscere meglio il Suo vicino

Here:

• tuo = your, informal singular
• Suo = your, formal singular

And if speaking to more than one person:

• il vostro vicino = your neighbor

Because meglio is an adverb modifying the verb phrase conoscere.

The natural order is:

• conoscere meglio qualcuno = to know someone better

So:

• Vorrei conoscere meglio il tuo vicino

This is the most natural word order. You might understand other word orders in context, but this one is the standard, smooth choice.

Yes. You can replace it with lo if it refers to a male neighbor.

So:

• Vorrei conoscere meglio il tuo vicino becomes
• Vorrei conoscerlo meglio

Notice that the pronoun is attached to the infinitive:

• conoscere + lo → conoscerlo

If it were a female neighbor:

• Vorrei conoscerla meglio

Yes, but it means something slightly different.

• Vorrei conoscere meglio il tuo vicino = I’d like to get to know your neighbor better
• Vorrei sapere di più sul tuo vicino = I’d like to know more about your neighbor

The first is about becoming more familiar with the person directly.
The second is about learning information about that person.

So both are correct, but they are not the same idea.