Мы хотим сыграть в футбол.

This is an aspect question.

• играть = imperfective
• сыграть = perfective

In this sentence, сыграть suggests playing one game/session from start to finish. It sounds like:

• we want to go play a game of soccer
• we want to have a soccer match

If you said Мы хотим играть в футбол, it would sound more like:

• we want to play soccer in general
• we want to be playing soccer
• we want to engage in the activity, without focusing on one completed game

So сыграть is natural when the idea is one occasion of playing.

The prefix helps form the perfective verb.

With играть, the action is seen as ongoing, repeated, or unbounded.

With сыграть, the action is seen as a whole event. In this context, it often means:

• to play a game
• to play one round/match
• to complete the act of playing

So с- here is not really something you translate separately as a word like with or from. It is part of the verb form that changes the aspect.

Russian uses играть/сыграть в with many games and sports:

• играть в футбол
• играть в теннис
• играть в шахматы

So в is just the normal preposition used with this expression.

Compare:

• играть в футбол = to play soccer
• играть на гитаре = to play the guitar

With games and sports, Russian usually uses в. With musical instruments, it usually uses на.

It is in the accusative case, because after играть в / сыграть в Russian uses the accusative.

Here, футбол looks exactly like the nominative because it is:

• masculine
• inanimate
• singular

For masculine inanimate nouns, the accusative singular is often the same as the nominative.

So:

• nominative: футбол
• accusative: футбол

They look the same, even though the grammatical case is different.

Хотим is the 1st person plural form of хотеть, meaning to want.

Conjugation of хотеть in the present tense:

• я хочу = I want
• ты хочешь = you want
• он/она хочет = he/she wants
• мы хотим = we want
• вы хотите = you want
• они хотят = they want

So мы хотим means we want.

In Russian, just like in English, want is normally in the present tense when you mean your current desire.

So:

• Мы хотим сыграть в футбол = we want to play soccer

The wanting is happening now, even if the playing happens later.

This is completely normal.

After хотеть, Russian usually uses an infinitive to say what someone wants to do.

Pattern:

• хотеть + infinitive

Examples:

• Я хочу спать = I want to sleep
• Она хочет читать = She wants to read
• Мы хотим сыграть в футбол = We want to play soccer

So сыграть is in the infinitive because it is the action that we want.

Yes, often you can.

Russian verbs usually show the subject clearly enough, so:

• Мы хотим сыграть в футбол
• Хотим сыграть в футбол

Both can mean We want to play soccer.

Including мы can make the subject clearer or more emphatic. Leaving it out can sound more natural in many contexts.

Yes. Russian word order is more flexible than English word order.

The neutral order here is:

• Мы хотим сыграть в футбол.

But depending on emphasis, you might also hear:

• В футбол мы хотим сыграть.
• Сыграть в футбол мы хотим.

These alternatives change the focus or emphasis, not the basic meaning.

For learners, the original order is the safest and most natural starting point.

A helpful approximation is:

• сыграть ≈ sih-GRAHT'

A more careful breakdown:

• сы- has the Russian vowel ы, which has no exact English equivalent
• stress is on -грать

Very rough pronunciation:

• [sɨˈgratʲ]

A few notes:

• ы is not the same as English ee
• the final ть is soft
• stress matters: сыгрАть

If pronunciation is your goal, ы is probably the hardest part of this sentence for an English speaker.

Usually it suggests one game/session.

Because of сыграть rather than играть, the sentence sounds like:

• we want to go play soccer
• we want to have a game of soccer
• we want to play a match

If you wanted to express a more general idea, Russian would more likely use играть:

• Мы хотим играть в футбол = we want to play soccer / we want to be playing soccer / we want to play soccer generally

So the aspect gives an important nuance.

Yes. In Russian, the standard expression is simply:

• играть в футбол

There is no article system like English a / the, so Russian does not need to decide between play soccer, play the soccer, or play a soccer game.

The noun just appears in the normal expression with в:

• в футбол
• в хоккей
• в теннис

So this is just the standard Russian way to talk about the sport.

Yes, and it has a slightly different feel.

• сыграть в футбол often suggests playing a game or match as a whole
• поиграть в футбол often suggests playing for a while, more casually

So:

• Мы хотим сыграть в футбол = we want to play a game/match of soccer
• Мы хотим поиграть в футбол = we want to play some soccer for a bit

In real conversation, both are common, but they are not exactly identical.