Learn Hardcore German: Am Wochenende besuchen wir unsere Großeltern im Dorf. - At the weekend we visit our grandparents in the village.

Literally it is on the weekend (British English) or on the weekend/over the weekend.
The article dem (dative, neuter, singular) is already inside am, so you must not add another article.

Why does the sentence start with Am Wochenende and put besuchen before wir?

German has a verb-second rule in main clauses: the conjugated verb must be the second element in the sentence.

You can put different elements in first position (subject, time, place, object, etc.). Here, the time phrase Am Wochenende is in first position.
The conjugated verb besuchen must therefore be second, and the subject wir comes after it:

Am Wochenende (1st element) besuchen (2nd element) wir …

This is completely normal word order in German.

Could I also say Wir besuchen am Wochenende unsere Großeltern im Dorf? Is it different in meaning?

Yes, that sentence is also correct:

Wir besuchen am Wochenende unsere Großeltern im Dorf.

The basic meaning is the same. The difference is mainly in emphasis:

Wir besuchen … → neutral, starts with the subject.
Am Wochenende besuchen wir … → puts extra emphasis on when it happens (the weekend).

Both are natural; you simply choose what you want to highlight.

Why is it unsere Großeltern and not unseren Großeltern?

Because unsere Großeltern is in the accusative plural as the direct object of besuchen.

The subject (who does something) is wir → nominative.
The thing/person we visit is unsere Großeltern → accusative object.

For the possessive unser in the plural:

Nominative plural: unsere Großeltern
Accusative plural: unsere Großeltern
Dative plural: unseren Großeltern

Here we need accusative, so unsere Großeltern is correct.
Unseren Großeltern would be dative, which you would use after certain prepositions like mit, zu, bei, etc., but not after besuchen.

What case is unsere Großeltern, and how can I tell?

unsere Großeltern is in the accusative case.

You can see it from the sentence structure:

wir = subject → nominative
besuchen = verb that takes a direct object
unsere Großeltern = direct object of besuchen → accusative
im Dorf = location → dative (because in with location uses dative)

The form unsere (plural) is the same in nominative and accusative, but the function in the sentence (object of besuchen) tells you it is accusative.

Why is there no preposition before unsere Großeltern? Could I say wir besuchen zu unseren Großeltern?

You must not add a preposition there. In German:

besuchen directly takes an accusative object (a person or place).
You say: wir besuchen unsere Großeltern, ich besuche meine Freunde, sie besucht die Schule, etc.

Wir besuchen zu unseren Großeltern is incorrect.
If you want to express motion towards them, you could say Wir fahren zu unseren Großeltern, but that is a different verb and a different structure. With besuchen, no preposition is used.

What does besuchen mean exactly, and how is it different from gehen zu or fahren zu?

besuchen means to visit someone or something (people, places, institutions):

Ich besuche meine Oma. – I visit my grandma.
Er besucht eine Sprachschule. – He attends a language school.

It focuses on being a guest or attending.

gehen zu and fahren zu focus on movement towards:

Ich gehe zu meiner Oma. – I go to my grandma(’s place).
Wir fahren zu unseren Großeltern. – We drive to our grandparents.

So:

besuchen = the act of visiting/being with someone.
gehen/fahren zu = the act of going/travelling to someone.

Why are Wochenende, Großeltern and Dorf capitalized?

They are all nouns, and in German all nouns are capitalized:

das Wochenende – the weekend
die Großeltern – the grandparents
das Dorf – the village

Words like am, wir, unsere, im stay lowercase because they are not nouns.

What is im in im Dorf, and why not say in dem Dorf?

im is a contraction of in dem:

in dem Dorf → im Dorf

Both are grammatically correct, but im is by far more common and sounds more natural in everyday language.
So im Dorf and in dem Dorf mean the same: in the village.

Is im Dorf dative or accusative, and why?

im Dorf is dative.

The preposition in is a two-way preposition. It takes:

dative for location (answering wo? – where?)
accusative for direction/movement (answering wohin? – where to?)

In the sentence, unsere Großeltern im Dorf tells us where the grandparents are (they are in the village), not where we are moving to. So it answers wo? and uses in + dem (dative) → im Dorf.

Could I say ins Dorf instead of im Dorf?

You can say ins Dorf, but it changes the meaning.

ins Dorf = in das Dorf → accusative → movement into the village (direction).
im Dorf = in dem Dorf → dative → in the village (location).

Compare:

Am Wochenende besuchen wir unsere Großeltern im Dorf.
→ Our grandparents are (live) in the village, and we visit them there.
Am Wochenende fahren wir ins Dorf.
→ We go/drive into the village (focus on movement to that place).

So im Dorf describes where the grandparents are; ins Dorf describes a movement toward the village.

What exactly does Großeltern mean? Is there a singular form?

Großeltern means grandparents and is always plural.

For individual grandparents, German normally uses:

die Großmutter – grandmother
der Großvater – grandfather
Colloquially: Oma, Opa

There is a technical word das Großelternteil (a grandparent, literally “grandparent part”), but it is rare in everyday speech. In normal conversation, people say eine Großmutter / ein Großvater.

Why is the verb form besuchen for wir the same as the infinitive?

For regular verbs like besuchen, the wir form in the present tense looks the same as the infinitive:

Infinitive: besuchen
wir: wir besuchen
sie (they): sie besuchen
Sie (formal you): Sie besuchen

You know it is a finite verb form here (not an infinitive) because:

it follows the verb-second rule in a main clause, and
there is an explicit subject wir.

So in this sentence, besuchen is present tense, 1st person plural.

The English translation might use the future (we’re going to visit). Why does German use the present tense here?

German often uses the present tense for future actions when there is a clear time expression:

Am Wochenende besuchen wir unsere Großeltern.
→ We will visit / We’re going to visit our grandparents at the weekend.

The time phrase Am Wochenende tells us it is in the future, so present tense is enough.
You could say Am Wochenende werden wir unsere Großeltern besuchen, but that is usually only needed for extra emphasis or clarity.

Can I make Wochenende plural, and why is it not am Wochenenden here?

Yes, Wochenende has a plural: die Wochenenden.

Singular dative: am Wochenende → on one specific weekend or “at the weekend” in general.
Plural dative: an den Wochenenden → on weekends (regularly, over several weekends).

Am Wochenenden is incorrect; you must also change the article:

an den Wochenenden (plural)
am Wochenende (singular or generic “at the weekend”)

In this sentence, we are talking about one coming weekend, so am Wochenende is the right form.