Learn Hardcore Polish: Wejście do muzeum jest naprzeciwko piekarni, więc kupimy bułki po wystawie. - The entrance to the museum is opposite the bakery, so we will buy rolls after the exhibition.

Because the preposition naprzeciwko (opposite/across from) takes the genitive case. The noun piekarnia (a bakery) in genitive singular is piekarni.
More examples:

naprzeciwko banku (bank → banku, gen. sg.)
naprzeciwko szkoły (szkoła → szkoły, gen. sg.)

What case is "muzeum" after "do", and why doesn’t it change form?

Do takes the genitive. Many neuter nouns ending in -um (muzeum, centrum, liceum) have the same form in nominative and genitive singular. So do muzeum is genitive even though it looks like the nominative. In the plural these nouns do change: e.g., nominative pl. muzea, genitive pl. muzeów.

What’s the difference between wejście and wstęp? And why do, not na?

wejście = the physical entrance or the act of entering (the doorway/entry point).
wstęp = admission/permission to enter.
With wejście, use:
do for entering a space/interior: wejście do muzeum (entrance to the museum building),
na for a surface/event: wejście na scenę (onto the stage), wejście na wystawę (admission to the exhibition).

Can I say naprzeciw instead of naprzeciwko? And what about przeciwko?

naprzeciwko and naprzeciw both mean “opposite/across from” and normally take the genitive (naprzeciw(ko) piekarni).
przeciwko means “against” (in opposition to) and takes the dative: przeciwko komu? czemu? (np. przeciwko planowi).

What does po mean here, and why is it wystawie (not wystawę)?

Here po means after (in time) and takes the locative case. wystawa → wystawie (locative sg.).
More time/sequence examples: po obiedzie (after lunch), po pracy (after work).

Could “po wystawie” be confused with “po bułki” (“for rolls”)?

It shouldn’t confuse native speakers because:

po meaning “after” takes the locative: po wystawie.
po meaning “for/to fetch” takes the accusative: po bułki.
Your sentence has the object first (bułki), then a time phrase (po wystawie), so it parses as “we’ll buy rolls, after the exhibition.” If you want zero ambiguity, you can front the time phrase: Więc po wystawie kupimy bułki.

Why is it kupimy and not kupujemy?

kupimy is the simple future of the perfective verb kupić and implies a single, completed action in the future (“we will buy”).
kupujemy is present (imperfective) and would mean “we are buying” (now) or, in some contexts, a scheduled future, but it’s less natural here.
If you wanted to emphasize a longer/ongoing activity in the future, use będziemy kupować (imperfective).

Can I move “po wystawie” elsewhere in the sentence?

Yes. Common options (all natural):

Wejście … naprzeciwko piekarni, więc po wystawie kupimy bułki.
Wejście … naprzeciwko piekarni, kupimy po wystawie bułki.
Wejście … naprzeciwko piekarni, bułki kupimy po wystawie (slightly emphasizing “bułki”).
Polish word order is flexible; moving the time phrase often just changes emphasis, not meaning.

Do I need a word for “some” before bułki?

No. Polish has no articles, and a bare plural often means “some/any.”

kupimy bułki = we’ll buy (some) rolls.
If you want to be more specific: kilka bułek (a few rolls), trochę bułek (some/a bit of rolls), jakieś bułki (some, unspecified).

How do I know which case po takes in general?

Rough guide:

po + locative = after (time), around/along/in (space), in (a language): po wystawie, po obiedzie; chodzić po parku; mówić po polsku.
po + accusative = for (to fetch/pick up/obtain): iść po bułki, po lekarza, po pomoc.

Is the comma before więc required? Are there alternatives to więc?

Yes, you normally put a comma before więc when it links clauses: …, więc …
Alternatives for “so/therefore”: dlatego (often with a comma or as a new sentence), zatem (more formal), w związku z tym (formal/longer).

Pronunciation tips for tricky words?

wejście [ˈvɛjɕ.t͡ɕɛ] – approx: VESH-chye (soft “ś/ć”).
naprzeciwko [naˈpʂɛt͡ɕifkɔ] – approx: nah-PSHEH-chif-koh (the “rz” is like “zh/ʐ”, “ci” before “w” is soft “ć”).
piekarni [pʲɛˈkarɲi] – approx: pye-KAR-nyi (soft “ń”).
wystawie [vɨˈstavjɛ] – approx: vih-STAH-vye (Polish “y” is like a hard “i”).
muzeum [muˈzɛum] – approx: moo-ZEH-oom.
Stress is almost always on the second-to-last syllable.

Is “naprzeciwko” one word?

Yes. It’s written as one word: naprzeciwko. Writing it as two words (na przeciwko) is a common mistake.

Could I also say “Wejście do muzeum znajduje się naprzeciwko piekarni”?

Yes. znajduje się (“is located”) sounds slightly more formal or descriptive, but it’s perfectly natural:

Wejście do muzeum jest/znajduje się naprzeciwko piekarni.
Both are correct.