Po pracy idę do kwiaciarni po bukiet dla babci.

Polish often leaves out subject pronouns when they are clear from the verb form.
Here, idę already means I am going / I go, so ja is unnecessary.

You could say Ja idę do kwiaciarni..., but that would usually add emphasis, like:

• Ja idę do kwiaciarni, nie on.
  I’m going to the flower shop, not him.

So the version without ja is the most natural neutral one.

No. It has two different meanings here:

• po pracy = after work
• po bukiet = for a bouquet / to get a bouquet

So in this sentence:

• Po pracy tells you when
• po bukiet tells you why / for what purpose

This is very normal in Polish. The meaning comes from the whole phrase, not just the preposition by itself.

Because Polish uses cases, and prepositions require different cases.

In this sentence:

• po pracy → pracy is after po meaning after, so it is in the locative
• do kwiaciarni → kwiaciarni is after do, so it is in the genitive
• dla babci → babci is after dla, so it is also in the genitive

So the endings are not random. They show the grammatical role required by each preposition.

Because do always requires the genitive case.

The dictionary form is:

• kwiaciarnia = flower shop

After do, it changes to:

• do kwiaciarni = to the flower shop

This is one of the most common patterns in Polish:

• do domu = to the house/home
• do szkoły = to school
• do sklepu = to the shop
• do kwiaciarni = to the flower shop

These verbs are related, but they are not interchangeable.

• idę = I’m going / I’m on my way / I’m going once now or as a current plan
• chodzę = I go regularly / habitually
• jadę = I’m going by vehicle
• pójdę = I will go

So:

• Po pracy idę do kwiaciarni... = After work, I’m going to the flower shop...
• Po pracy chodzę do kwiaciarni would suggest a repeated habit
• Po pracy jadę do kwiaciarni would suggest going by car, bus, etc.
• Po pracy pójdę do kwiaciarni means I will go after work

iść po coś is a very common Polish pattern meaning to go get something, to go pick something up, or to go buy something.

So:

• idę po bukiet = I’m going to get a bouquet

It does not mean the same as English after a bouquet. It is an idiomatic Polish structure.

Other examples:

• Idę po chleb. = I’m going to get bread.
• Poszedł po lekarza. = He went to get a doctor.

A useful detail: in po bukiet, the noun is in the accusative, but bukiet looks the same as the basic form, so you do not see a change here.

If you want to say the bouquet is for grandma, Polish normally uses dla + genitive:

• bukiet dla babci = a bouquet for grandma

You cannot just say po bukiet babci to mean that here. That would sound wrong or suggest a different relationship.

Compare:

• Mam bukiet dla babci. = I have a bouquet for grandma.
• Daję babci bukiet. = I’m giving grandma a bouquet.

In the second sentence, babci appears without dla, but that is because it is the indirect object of the verb dawać. In your original sentence, the noun phrase is bukiet dla babci, so dla is the natural choice.

Polish word order is fairly flexible, but the original order sounds very natural and neutral:

• Po pracy idę do kwiaciarni po bukiet dla babci.

It flows like this:

1. Po pracy = time
2. idę do kwiaciarni = action + destination
3. po bukiet dla babci = purpose

You can move parts around, for example:

• Idę po pracy do kwiaciarni po bukiet dla babci.
• Po pracy idę po bukiet dla babci do kwiaciarni.

These are understandable, but they may shift emphasis or sound a little less neutral. For a learner, the original version is an excellent standard model.

Polish does not have articles like English a/an and the.

So:

• bukiet can mean a bouquet or the bouquet
• kwiaciarnia can mean a flower shop or the flower shop

The exact meaning depends on context. In this sentence, English usually translates it naturally as:

• After work I’m going to the flower shop to get a bouquet for my grandma.
• or ...to a flower shop...

Polish simply does not need articles to express that.