אני רוצה להכיר את המדינה הזאת טוב יותר.

Here, להכיר means to get to know or to become familiar with.

With people, places, cities, and countries, Hebrew usually uses להכיר for the idea of being acquainted or becoming familiar with something.

• להכיר אדם = to know / get to know a person
• להכיר עיר = to know / get to know a city
• להכיר מדינה = to know / get to know a country

With people, להכיר can also mean to meet in the sense of first getting acquainted, as in נעים להכיר = nice to meet you. But in this sentence, with country, it clearly means become more familiar with, not meet.

Because להכיר and לדעת are not the same kind of know.

• להכיר = to know by acquaintance / to be familiar with / to get to know
• לדעת = to know a fact, piece of information, or how to do something

So:

• אני מכיר את האיש הזה = I know this man
• אני מכיר את העיר הזאת = I know this city
• אני יודע את התשובה = I know the answer
• אני יודע לשחות = I know how to swim

A country is something you become familiar with, so להכיר is the natural choice.

This is a very common Hebrew structure:

• רוצה = want / wants
• להכיר = to get to know

Together, רוצה להכיר means want to get to know.

Hebrew often uses: [present-tense verb] + infinitive

So just as English says want to know, Hebrew says רוצה להכיר.

The ל־ at the start of להכיר is the normal infinitive marker, similar to English to in to know.

In normal unpointed Hebrew spelling, it would usually look exactly the same:

אני רוצה להכיר את המדינה הזאת טוב יותר.

But the pronunciation changes:

• male speaker: ani rotze
• female speaker: ani rotza

So the word רוצה is spelled the same here, but pronounced differently depending on gender.

With vowel marks, they are different:

• masculine: רוֹצֶה
• feminine: רוֹצָה

את marks a definite direct object.

It does not mean with here. It has no direct English equivalent in this sentence.

You use את before a direct object when that object is definite, for example:

• את המדינה הזאת = this country
• את הספר = the book
• את דן = Dan

So in this sentence:

• אני רוצה להכיר את המדינה הזאת
  = I want to get to know this country

If the object were indefinite, you would usually not use את:

• אני רוצה להכיר מדינה חדשה
  = I want to get to know a new country

When this directly modifies a noun in Hebrew, it usually comes after the noun:

• הספר הזה = this book
• העיר הזאת = this city
• המדינה הזאת = this country

So the normal pattern is:

the + noun + this

By contrast, זאת המדינה is usually a different kind of structure, more like:

• This is the country
• or an emphatic this country in a special context

For the basic phrase this country, המדינה הזאת is the standard form.

Because in Hebrew, nouns with demonstratives like this and that are normally marked as definite on the noun too.

So Hebrew says:

• המדינה הזאת = this country
• הספר הזה = this book
• הילד הזה = this boy

To an English speaker, this can feel like the country this, but that is just the normal Hebrew pattern.

So yes, both parts help show definiteness:

• המדינה
• הזאת

Yes.

Both המדינה הזאת and המדינה הזו mean this country.

A rough difference:

• הזאת = fuller form, often felt as slightly more formal or careful
• הזו = shorter form, very common in everyday use

Both are correct and natural.

Also note:

• זו המדינה usually means this is the country
• after a noun, you normally want הזו or הזאת

Because מדינה is a feminine singular noun, and the demonstrative must agree with it.

Agreement here is by gender and number:

• masculine singular: הזה
• feminine singular: הזאת / הזו
• plural: האלה

Examples:

• הספר הזה = this book
• המדינה הזאת = this country
• הערים האלה = these cities

So הזאת is used because מדינה is feminine.

טוב יותר means better.

Literally, it is close to more well or better. In this sentence, it describes how well the speaker wants to know the country.

That is why it is טוב, not טובה: it is not describing the country itself. It is describing the action להכיר.

So:

• להכיר את המדינה הזאת טוב יותר
  = to know this country better

But:

• המדינה הזאת טובה יותר
  = this country is better

In the second sentence, טובה agrees with מדינה, because now it is describing the country.

Also, in everyday Hebrew, many speakers would say יותר טוב instead of טוב יותר. Both mean better.
A simple distinction is:

• טוב יותר = a bit more formal / standard
• יותר טוב = very common in speech

Often yes, especially if the subject is already clear from context.

So you may hear:

רוצה להכיר את המדינה הזאת טוב יותר.

However, keeping אני is very natural, especially when the sentence stands alone.

One reason is that Hebrew present-tense forms do not clearly show person the way English does.
For example, רוצה can mean:

• I want
• you want
• he wants

depending on context and gender.

So אני helps make the subject clear.