Moja siostra urodziła się szóstego czerwca, a nasz wyjazd nad morze zaczyna się siódmego sierpnia.

In Polish, urodzić się means to be born.

So:

• Moja siostra urodziła się = My sister was born

Without się, urodzić usually means to give birth to.

For example:

• Ona urodziła dziecko = She gave birth to a child
• Ona urodziła się w czerwcu = She was born in June

So in this sentence, się is necessary because the meaning is was born.

Because siostra is feminine.

In the past tense, Polish verbs agree with the gender of the subject:

• urodził się = masculine
• urodziła się = feminine
• urodziło się = neuter

So:

• brat urodził się... = brother was born...
• siostra urodziła się... = sister was born...

Because Polish normally expresses calendar dates with:

• an ordinal number in the genitive
• the month name also in the genitive

So:

• szósty → szóstego
• siódmy → siódmego
• czerwiec → czerwca
• sierpień → sierpnia

This is the normal pattern for dates such as:

• pierwszego maja = on the first of May
• dwudziestego lipca = on the twentieth of July

So szóstego czerwca literally corresponds to the sixth of June.

Usually, no preposition is needed when giving a date like this.

So Polish says:

• Urodziła się szóstego czerwca
• literally: She was born sixth of June
• naturally: She was born on the sixth of June

English needs on, but Polish often does not.

You may sometimes see:

• dnia szóstego czerwca

This is more formal. In everyday speech, just szóstego czerwca is normal.

Because possessive words like my and our must agree with the noun they describe.

• siostra is feminine, so: moja siostra
• wyjazd is masculine, so: nasz wyjazd

Compare:

• mój brat = my brother
• moja siostra = my sister
• moje dziecko = my child

And:

• nasz dom = our house
• nasza szkoła = our school
• nasze auto = our car

So the form changes depending on the gender of the noun.

Wyjazd means something like:

• trip
• departure
• going away
• getaway

In this sentence, nasz wyjazd nad morze means our trip to the seaside or our seaside trip.

It is not just the abstract act of leaving. In many contexts, wyjazd refers to the whole trip, especially when you are going somewhere for a holiday or a stay.

Because nad changes case depending on movement vs. location.

• nad + accusative = movement to/towards
• nad + instrumental = location at/by/over

So:

• jedziemy nad morze = we’re going to the seaside
• jesteśmy nad morzem = we are at the seaside

In your sentence, wyjazd nad morze involves movement to a destination, so Polish uses the accusative:

• morze = accusative form here

Because nad morze means going to the area by the sea, not literally into the sea or to the sea as an object.

In Polish, jechać nad morze is the standard expression for to go to the seaside.

Compare:

• jadę nad morze = I’m going to the seaside
• mieszkam nad morzem = I live by the sea

Using do morza would usually sound wrong in this context. Polish treats the seaside as a place you go nad morze, just as you go:

• w góry = to the mountains
• na wieś = to the countryside

Here, zaczyna się means begins / starts.

The verb is often used with się when something begins on its own, or when we present it that way:

• Film zaczyna się o ósmej = The film starts at eight
• Rok szkolny zaczyna się we wrześniu = The school year begins in September

So:

• nasz wyjazd ... zaczyna się siódmego sierpnia = our trip ... begins on the seventh of August

Without się, zaczynać is more often used with a direct object:

• zaczynam lekcję = I am starting the lesson
• zaczynam pracę = I am starting work

Because a is normally preceded by a comma when it joins two clauses.

In this sentence, it connects:

• Moja siostra urodziła się szóstego czerwca
• nasz wyjazd nad morze zaczyna się siódmego sierpnia

So the comma is standard Polish punctuation here.

Also, a often has a slight contrastive or linking feel, not just a simple and. It can suggest something like and meanwhile, while, or whereas, depending on context.

Both can often be translated as and, but they are not always used in the same way.

• i simply adds one thing to another
• a often links two separate but related facts, sometimes with a mild contrast

In your sentence, a sounds natural because it connects two different facts:

• my sister’s birth date
• the start date of our trip

So a is a very natural choice here.

If you used i, it would sound more like a plain addition. Not impossible in every context, but a fits better in this sentence.

Yes, Polish word order is fairly flexible, although some versions sound more natural than others.

For example, you could say:

• Moja siostra urodziła się szóstego czerwca, a nasz wyjazd nad morze zaczyna się siódmego sierpnia.
• Szóstego czerwca urodziła się moja siostra, a siódmego sierpnia zaczyna się nasz wyjazd nad morze.

Both are grammatical, but the emphasis changes.

The original version is neutral and natural. Moving the dates to the front gives them more emphasis.

A very common question is:

• Kiedy urodziła się twoja siostra? = When was your sister born?
• Kiedy zaczyna się wasz wyjazd nad morze? = When does your trip to the seaside start?

And the answers could be:

• Szóstego czerwca.
• Siódmego sierpnia.

So the same date form is used both in full sentences and in short answers.