Po obiedzie mam zamiar kupić lody, gruszki i śliwki na deser.

Because po meaning after takes the locative case in this time expression.

• basic form: obiad
• after po: po obiedzie

So po obiedzie means after lunch / after dinner.
This is a very common pattern in Polish:

• po pracy = after work
• po szkole = after school
• po obiedzie = after lunch/dinner

Obiad is one of those Polish meal words that does not match English perfectly.

In modern Polish, obiad usually means the main cooked meal of the day. Depending on the person, family, or schedule, that may be closer to:

• lunch
• midday dinner
• sometimes just dinner

So if you are translating into English, the best choice depends on context. The important thing in Polish is that obiad is the main substantial meal.

Mam zamiar kupić means I intend to buy or I’m planning to buy. It emphasizes intention.

Compare:

• Po obiedzie mam zamiar kupić... = After lunch, I intend to buy...
• Po obiedzie kupię... = After lunch, I will buy...

Both are correct, but the version with mam zamiar sounds a bit more like the speaker is talking about a plan.

This is about aspect, which is very important in Polish.

• kupić = perfective = to buy, with the action seen as completed
• kupować = imperfective = to be buying / to buy in a general or repeated sense

After mam zamiar, Polish often uses the infinitive that matches what you mean. Here, the speaker means one completed shopping action, so kupić is the natural choice.

So:

• mam zamiar kupić lody = I intend to buy ice cream
• mam zamiar kupować lody could suggest buying ice cream regularly or as an ongoing activity, which is not the idea here

Yes, and this is a very common surprise for English speakers.

In Polish, lody is a plural-only noun in ordinary usage. So Polish treats ice cream as grammatically plural.

You say:

• lody są dobre = the ice cream is good
• jem lody = I’m eating ice cream

There is no normal everyday singular lód with the meaning one ice cream in this context.
The word lód exists, but it means ice, not an ice cream.

They are in the accusative case because they are the direct objects of kupić.

The speaker is buying what?

• lody
• gruszki
• śliwki

That means these nouns are objects, so they go in the accusative.

In this sentence, the forms happen to look the same as the nominative plural:

• lody
• gruszki
• śliwki

That is normal for many non-masculine-personal plural nouns in Polish.

Yes, both are plural.

Singular and plural:

• gruszka = a pear

• gruszki = pears

• śliwka = a plum
• śliwki = plums

So the sentence is talking about buying multiple pears and multiple plums.

Na deser means for dessert.

Here, na is used in a very common Polish pattern meaning for or as something. With this meaning, na takes the accusative case.

So:

• basic form: deser
• accusative singular: deser
  It looks the same here because deser is an inanimate masculine noun.

Other similar examples:

• na prezent = as a gift / for a present
• na śniadanie = for breakfast
• na kolację = for dinner/supper

So na deser is simply the normal idiomatic way to say for dessert.

Because i means and, and in a simple list Polish normally does not put a comma before it.

So:

• lody, gruszki i śliwki

This works just like standard English without the Oxford comma.

Polish punctuation usually avoids a comma before i unless the structure is more complicated and a separate rule requires one.

Yes. Polish word order is more flexible than English word order.

The sentence as given is neutral and natural:

• Po obiedzie mam zamiar kupić lody, gruszki i śliwki na deser.

But you could move parts around to change emphasis:

• Mam zamiar kupić po obiedzie lody, gruszki i śliwki na deser.
• Lody, gruszki i śliwki mam zamiar kupić po obiedzie na deser.

These alternatives are grammatical, but they may sound more marked or emphasize different parts. For a learner, the original order is a very good default.

Because Polish has no articles.

English needs words like:

• a pear
• the pears
• some ice cream

Polish usually does not. The meaning is understood from context.

So:

• kupić gruszki can mean buy pears, buy the pears, or sometimes buy some pears, depending on the situation

This is normal in Polish and something English speakers have to get used to.

A few parts may feel unusual for an English speaker:

• kupić
  The final ć is a soft sound, a bit like a very soft ch.

• lody
  The ł is pronounced like English w, so it sounds roughly like LO-dy with a w-like sound in other words containing ł. Here there is no ł, just plain l, so it is simpler.

• gruszki
  sz sounds like English sh.
  So roughly: GROOSH-kee

• śliwki
  ś is a softer sound than English sh.
  Roughly: SHLEEF-kee, but softer at the start.

• obiedzie
  dzie is a soft sound, somewhat like jye or dye said softly.

Also, Polish usually stresses the second-to-last syllable of a word, so:

• o-BIE-dzie
• ZA-miar
• KU-pić
• GRUSZ-ki
• ŚLIW-ki
• DE-ser

It is completely normal everyday Polish.

It is neutral and common in speech and writing. It means something like:

• I intend to
• I plan to
• I mean to

Other possibilities are:

• chcę kupić = I want to buy
• zamierzam kupić = I intend to buy
• kupię = I’ll buy

But mam zamiar kupić is very natural and useful to know.