V kavárně si vezmu kávu bez cukru.

Because v meaning in is followed by the locative case when it shows location.

• dictionary form: kavárna = café
• locative singular: v kavárně = in the café

So this is a normal case change after the preposition v.

Because kávu is the direct object of the verb vezmu.

The noun káva is feminine, and in the singular its accusative form is:

• nominative: káva
• accusative: kávu

Since the speaker is taking/ordering a coffee, Czech puts káva into the accusative:

• vezmu kávu = I’ll take / have a coffee

Si is a very common Czech clitic. In this sentence it means something like for myself.

So:

• vezmu kávu = I’ll take a coffee
• vezmu si kávu = I’ll take/get myself a coffee

In English we usually do not translate si directly, but in Czech it often appears with actions people do for their own benefit, especially with food and drinks.

This is very natural in sentences like:

• Dám si kávu. = I’ll have a coffee.
• Vezmu si kávu. = I’ll take/have a coffee.

Yes, the form looks present, but vzít is a perfective verb. In Czech, perfective verbs do not have a true present meaning. Their “present” forms usually refer to the future.

So:

• vezmu = I will take / I’ll have

This is why vezmu can be translated as a future idea in English.

Compare:

• beru = I am taking / I take (imperfective)
• vezmu = I will take / I’ll take (perfective)

Because the preposition bez (without) requires the genitive case.

• dictionary form: cukr = sugar
• genitive singular: cukru

So:

• bez cukru = without sugar

This is a very useful pattern to remember:

• bez mléka = without milk
• bez problémů = without problems
• bez cukru = without sugar

It is understood as describing the coffee:

• kávu bez cukru = coffee without sugar

So the structure is basically:

• V kavárně = in the café
• si vezmu = I’ll have / take for myself
• kávu bez cukru = coffee without sugar

Because Czech has no articles.

So Czech simply says:

• kavárna = café / a café / the café
• káva = coffee / a coffee / the coffee

English needs articles, but Czech usually leaves that information to context.

No, Czech word order is fairly flexible, but this version sounds natural and neutral.

This sentence starts with V kavárně to set the scene: In the café...

Then comes si vezmu, and then the thing being taken:

• kávu bez cukru

Other word orders are possible, but they change emphasis. For example:

• Kávu bez cukru si vezmu v kavárně.
  This puts more focus on coffee without sugar.

The original sentence is a normal, natural way to say it.

Because si is a clitic, and Czech clitics usually go in the second position in the clause.

So in:

• V kavárně si vezmu kávu bez cukru.

the first unit is V kavárně, and si comes right after it.

This second-position behavior is very common with words like:

• si
• se
• mi
• ti
• ho
• jsem

So even though English learners may want to place si right next to the verb, Czech often puts it in second position instead.

Yes, absolutely. In everyday Czech, Dám si kávu bez cukru is probably even more common when ordering or deciding what to have.

Both are possible:

• Dám si kávu bez cukru. = I’ll have a coffee without sugar.
• Vezmu si kávu bez cukru. = I’ll take/have a coffee without sugar.

Dám si often sounds especially natural for food and drinks.

The infinitive is vzít = to take.

Some important forms are:

• vzít = to take
• vezmu = I will take
• vezmeš = you will take
• vezme = he/she/it will take

The verb is a bit irregular, so it is worth learning as a full pattern rather than trying to build it from the infinitive mechanically.

They would hear it as a very normal statement like:

• In the café, I’ll have a coffee without sugar.

The sentence sounds natural and idiomatic because it uses:

• v + locative for location
• si for a personal, self-benefiting action
• vezmu as a perfective future
• accusative for the thing being taken
• bez + genitive for without sugar

So it is a good example of several very common Czech patterns working together.