Siódmego stycznia jedziemy do dziadków na kolację.

Because Polish dates are usually expressed with:

• an ordinal number for the day
• the month in the genitive case

So:

• siódmy = seventh (basic dictionary form)
• siódmego = genitive form, used in dates
• styczeń = January
• stycznia = genitive form of styczeń

So siódmego stycznia means on the seventh of January / January 7th.

You can think of it as coming from an older fuller structure like siódmego dnia stycznia (of the seventh day of January), even though dnia is normally omitted.

In Polish, dates often appear without a preposition. A bare date in this form can mean on January 7th.

So:

• Siódmego stycznia jedziemy... = On January 7th, we’re going...

You may also sometimes hear w styczniu (in January) for the month in general, but for a specific date Polish commonly uses just the date itself, without w.

Because stycznia is the genitive singular of styczeń.

Months often change form depending on grammar:

• styczeń = January
• w styczniu = in January
• do stycznia = until January
• siódmego stycznia = on the 7th of January

So this is a normal case change, not a different word.

Polish distinguishes between:

• iść / idziemy = to go on foot
• jechać / jedziemy = to go by vehicle or to travel

So jedziemy do dziadków suggests we are going to our grandparents’ place by car, bus, train, etc.

If the speakers were walking, you would expect:

• Idziemy do dziadków na kolację.

Because in Polish, a present-tense form of an imperfective verb can be used for a planned or scheduled future action, especially when the time is clear from context.

Here, Siódmego stycznia makes the future time obvious, so:

• jedziemy literally looks present
• but in context it means we’re going / we’ll be going

This is very similar to English We’re going to Grandma and Grandpa’s for dinner on January 7th.

Because do means to in the sense of movement toward a place/person, and do requires the genitive case.

So:

• dziadkowie = grandparents
• dziadków = genitive form

Thus:

• do dziadków = to the grandparents / to our grandparents’ place

In Polish, do dziadków often really means to Grandma and Grandpa’s house, not just physically toward the people.

Not in this sentence. Here it comes from dziadkowie, which means grandparents.

Important distinction:

• dziadek = grandfather
• dziadkowie = grandparents
• dziadków can be:
  • genitive plural of dziadkowie (of/to grandparents)
  • or accusative plural/genitive plural of dziadek in other contexts

Here, because of meaning and structure, do dziadków clearly means to (our) grandparents.

Because na is commonly used with meals, events, and purposes of going somewhere.

So:

• na kolację = for dinner / for supper
• iść / jechać na kolację = to go for dinner

The preposition na here takes the accusative case, and:

• kolacja → kolację

So the structure means something like:

• We’re going to our grandparents’ place for dinner.

It usually means the evening meal, so depending on the variety of English it may be translated as:

• dinner
• supper

In many learning materials, kolacja is often glossed as supper, while obiad is usually the main meal earlier in the day and is often translated as lunch or dinner, depending on context.

So in this sentence, na kolację is best understood as for the evening meal.

Yes. Polish word order is fairly flexible because cases show grammatical relationships.

This sentence:

• Siódmego stycznia jedziemy do dziadków na kolację.

is a very natural neutral order.

But other orders are also possible, for example:

• Jedziemy do dziadków na kolację siódmego stycznia.
• Do dziadków jedziemy siódmego stycznia na kolację.

These variations can change emphasis slightly, but the basic meaning stays the same.

Siódmego is capitalized only because it is the first word of the sentence.

In Polish:

• days of the week are normally lowercase
• months are normally lowercase

So stycznia stays lowercase.

If the date appeared in the middle of a sentence, you would normally write:

• Jedziemy siódmego stycznia do dziadków na kolację.