У нас общая цель — говорить по-русски свободно.

In Russian, у + genitive is a very common way to express having something.

So У нас общая цель literally looks like At us, there is a shared goal, but in natural English it means We have a shared goal.

Russian often prefers this structure over a verb meaning to have.

• У нас есть план. = We have a plan.
• У меня есть вопрос. = I have a question.

In your sentence, есть is omitted because in the present tense it is often left out when the meaning is clear:

• У нас общая цель. = We have a shared goal.

Нас is in the genitive case.

That is because the preposition у requires the genitive:

• у меня
• у тебя
• у нас
• у них

So:

• у = at/by
• нас = genitive form of мы

Together, у нас means something like with us / in our possession / on our side, depending on context. Here it gives the idea we have.

Because цель is a feminine noun, and the adjective has to agree with it in gender, number, and case.

• цель = feminine singular
• so the adjective must also be feminine singular: общая

Compare:

• общий план = a shared plan, a masculine noun
• общая цель = a shared goal, a feminine noun
• общее дело = a shared/common matter, a neuter noun

This is standard adjective-noun agreement in Russian.

Общая here means shared, common, or the same for all of us.

So общая цель is not just any goal. It means a goal that we all share together.

Depending on context, it can be translated as:

• a common goal
• a shared goal

Russian often uses a dash between two parts of a sentence when one part identifies or defines the other, especially in more formal or clear written style.

Here:

• У нас общая цель = We have a shared goal
• говорить по-русски свободно = what that goal is

So the dash works like namely / that is / which is.

A natural English version would be:

• We have a common goal — to speak Russian fluently.

You could think of the dash as marking an explanation of what the goal is.

Because after цель (goal), Russian often uses an infinitive to say what the goal is.

So:

• цель — говорить...
• literally: the goal is to speak...

This is very similar to English:

• Our goal is to speak Russian fluently.

Other examples:

• Моя цель — работать за границей. = My goal is to work abroad.
• Наша цель — закончить проект. = Our goal is to finish the project.

Because in Russian, when you say to speak a language, you usually use either:

• говорить по-русски = to speak Russian
• говорить на русском (языке) = to speak in Russian / to speak the Russian language

But говорить русский is not the normal pattern.

The adverb по-русски means in Russian / Russian-style, and with говорить it naturally means to speak Russian.

Common patterns:

• говорить по-английски = speak English
• говорить по-французски = speak French
• говорить по-русски = speak Russian

Both can mean in Russian, but they are used a little differently.

по-русски

Usually emphasizes the way of speaking or simply means Russian as a language adverb.

• Я говорю по-русски. = I speak Russian.

на русском

Literally in Russian, often used when talking about the language of communication, text, media, or an activity.

• Книга на русском. = The book is in Russian.
• Мы говорим на русском. = We speak in Russian.

In your sentence, говорить по-русски is the most natural and standard way to say to speak Russian.

Свободно is an adverb because it describes how someone speaks.

• говорить = to speak
• свободно = freely / fluently

Here it means fluently, not literally in a free way.

So:

• говорить по-русски свободно = to speak Russian fluently

Russian often uses свободно where English uses fluently:

• Он свободно говорит по-испански. = He speaks Spanish fluently.

Yes, word order can vary.

These are all possible:

• говорить по-русски свободно
• свободно говорить по-русски
• говорить свободно по-русски

But they do not all sound equally neutral in every context.

In your sentence, говорить по-русски свободно sounds natural and clear.
A very common alternative is:

• свободно говорить по-русски

That version may feel slightly more natural to some speakers because свободно stays close to говорить.

Russian word order is flexible, but changing it can slightly shift emphasis.

No, Russian word order is fairly flexible, but the original sentence is natural and neutral.

Possible variations include:

• Наша общая цель — свободно говорить по-русски.
• Общая цель у нас — говорить по-русски свободно.

These are grammatically possible, but they may change the emphasis:

• У нас общая цель... focuses first on we/us
• Наша общая цель... sounds a bit more direct: our shared goal
• Общая цель у нас... can sound more contrastive or stylistically marked

So the original is a good standard version.

Yes, absolutely.

• У нас общая цель... = We have a shared goal...
• Наша общая цель... = Our shared goal...

Both are natural.

The difference is mostly in structure and emphasis:

• У нас общая цель highlights possession: we have
• Наша общая цель directly names the goal as ours

In many contexts, they are very close in meaning.

The main stresses are:

• У нас общая це́ль — говори́ть по-ру́сски свобо́дно.

A few useful notes:

• це́ль = one syllable with stressed е
• говори́ть = stress on the last syllable
• по-ру́сски = stress on ру
• свобо́дно = stress on бо

Stress is important in Russian, so it is worth learning words together with their accent.

Very often, yes, when you want to say what the goal is.

Examples:

• Цель — победить. = The goal is to win.
• Моя цель — выучить русский. = My goal is to learn Russian.

But цель can also be followed by other structures, depending on meaning:

• цель проекта = the goal of the project
• с целью обучения = for the purpose of study

So the infinitive is very common, but not the only possibility.