Learn Hardcore Finnish: Ensimmäinen pysähdys on kirjaston edessä, toinen uimahallin takana. - The first stop is in front of the library, the second behind the swimming hall.

pysäkki = a bus/tram stop, the physical stop itself (the sign, the place on the route).
pysähdys = a stopping, an act or instance of stopping; also used for “stop” in a more general sense (e.g. stops on a tour, a trip, etc.).

In a context like “the first stop is in front of the library”, either can be used depending on nuance:

Ensimmäinen pysäkki on kirjaston edessä.
→ The first (bus) stop is in front of the library. (emphasis on the official bus stop location)
Ensimmäinen pysähdys on kirjaston edessä.
→ The first stop is in front of the library. (emphasis on the event of stopping; could be a sightseeing stop, a break on a trip, etc.)

So the sentence you have sounds a bit more like describing stops on some route or tour, not necessarily just the fixed bus stops on a timetable.

Why do kirjaston and uimahallin end in -n?

The -n ending here is the genitive case in Finnish.

kirjasto = library
→ kirjaston = of the library / the library’s
uimahalli = swimming hall / swimming pool building
→ uimahallin = of the swimming hall

Some words like edessä (“in front of”) and takana (“behind”) work like postpositions: they need the preceding noun in the genitive:

kirjaston edessä = in front of the library (literally “at the front of the library”)
uimahallin takana = behind the swimming hall (literally “at the back of the swimming hall”)

So the -n doesn’t mean possession in the English sense here; it’s just required by the postpositions edessä and takana.

Why is it kirjaston edessä and not something like edessä kirjasto, like an English preposition?

Finnish mostly uses postpositions, which come after the noun, not before it:

English: in front of the library
Finnish structure: library’s in-front → kirjaston edessä

Pattern:

Noun in genitive (kirjaston, uimahallin)
Postposition (edessä, takana, lähellä, vieressä, etc.)

So:

kirjaston edessä = in front of the library
uimahallin takana = behind the swimming hall
kaupan vieressä = next to the shop
koulun lähellä = near the school

If you reverse the order (edessä kirjasto), it’s simply wrong in standard Finnish.

What exactly are edessä and takana? Are they cases, prepositions, or something else?

In practice, you can think of edessä and takana as postpositions that:

describe a location (in front of, behind)
require the genitive form of the noun before them

Usage:

kirjaston edessä = in front of the library
uimahallin takana = behind the swimming hall

Historically they come from nouns plus case endings (e.g. ete- + adessive -ssä, taka + adessive -na), but for a learner it’s simpler to just memorize them as postposition words:

edessä – in front (of)
takana – behind
alla – under
päällä – on (top of)
välissä – between

and remember: they go after a genitive noun.

Why is the verb on missing in the second part: “..., toinen uimahallin takana”? Shouldn’t it be toinen on uimahallin takana?

The full, “complete” sentence would indeed be:

Ensimmäinen pysähdys on kirjaston edessä, toinen on uimahallin takana.

But in Finnish, when two clauses share the same verb, it’s very common to omit the repeated verb in the second clause, especially in short, clear sentences:

Minä olen opettaja, sinä (olet) opiskelija.
Tämä on helppoa, tuo (on) vaikeaa.

So your sentence is just a natural, slightly shorter version:

Ensimmäinen pysähdys on kirjaston edessä, toinen (on) uimahallin takana.

The verb on is understood from context and doesn’t need to be repeated.

Why doesn’t the second part repeat pysähdys? Why just toinen uimahallin takana?

This is a case of ellipsis: leaving out words that are obvious from context.

First part:

Ensimmäinen pysähdys on kirjaston edessä
→ The first stop is in front of the library.

Second part:

(Toinen pysähdys on) uimahallin takana.

Because pysähdys and on are already given in the first clause, Finnish can comfortably leave them out in the second, as long as the meaning is clear. A more “fully spelled out” version would be:

Ensimmäinen pysähdys on kirjaston edessä, toinen pysähdys on uimahallin takana.

Your original sentence is just more natural and less repetitive.

What’s the difference between ensimmäinen / toinen and the basic numbers yksi / kaksi?

yksi, kaksi, kolme, ... are cardinal numbers (one, two, three…).
ensimmäinen, toinen, kolmas, ... are ordinal numbers (first, second, third…).

In English:

one stop vs. the first stop

In Finnish:

yksi pysähdys = one stop
ensimmäinen pysähdys = the first stop

The key forms here:

ensimmäinen = first
toinen = second
kolmas = third

In your sentence:

Ensimmäinen pysähdys = the first stop
toinen (pysähdys) = the second (stop)

How would I say “The first stop is at the library” instead of “in front of the library”?

You would change the location expression:

kirjaston edessä = in front of the library
kirjastossa = in/at the library (inside the building or at the library as a place)

If you want “at the library” in the general sense (the stop is located at the library), you can say:

Ensimmäinen pysähdys on kirjastossa.
→ The first stop is at the library / in the library area.

Compare:

Ensimmäinen pysähdys on kirjaston edessä.
→ The first stop is in front of the library (physically in front of the building).

So:

kirjastossa: inessive case (-ssa/-ssä) = in/at the library
kirjaston edessä: genitive + postposition = in front of the library

How would this sentence look in the plural, e.g. “The first stops are in front of the library, the second ones behind the swimming hall”?

You’d need plural forms for “first” and “stop” and plural verb agreement:

Ensimmäiset pysähdykset ovat kirjaston edessä, toiset uimahallin takana.

Breakdown:

ensimmäiset = first (plural: “the first ones”)
pysähdykset = stops (plural nominative)
ovat = are (3rd person plural of olla)
toiset = the second ones (plural)

You can still omit the repeated words in the second clause:

Fuller: Ensimmäiset pysähdykset ovat kirjaston edessä, toiset pysähdykset ovat uimahallin takana.
Natural: Ensimmäiset pysähdykset ovat kirjaston edessä, toiset uimahallin takana.

Why isn’t there any word for “the” in “the first stop”? How do I know it’s “the” and not “a”?

Finnish has no articles at all—no equivalent of a/an or the.

The noun phrase ensimmäinen pysähdys can mean:

the first stop
a first stop
depending on context.

In practice:

When the phrase is clearly referring to a specific, known stop on a route (like in your sentence), it’s naturally understood as “the first stop”.
If you needed to stress that it’s not just any first stop but a specific, previously mentioned one, you’d rely on context, word order, or additional words (like juuri = “exactly / just”), not an article.

So you don’t translate “the” or “a” into Finnish; you infer that from context when going back into English.