Learn Hardcore German: Manchmal fahre ich nach der Arbeit mit dem Roller zum Spielplatz, um das Baby kurz zu sehen. - Sometimes I go by scooter to the playground after work in order to see the baby for a moment.

The finite verb (here: fahre) must always be in the second position.
The first position can be the subject, an adverb, a time expression, etc.

So:

Manchmal fahre ich …
1st position: Manchmal (adverb)
2nd position: fahre (verb)
Then: ich …

You can indeed also say:

Ich fahre manchmal nach der Arbeit mit dem Roller zum Spielplatz, um das Baby kurz zu sehen.

That version is equally correct. Putting Manchmal at the beginning simply emphasizes “Sometimes (as opposed to always)” a bit more.

Why is it ich fahre and not ich gehe? In English I might say “I go to the playground…”.

In German, fahren is used specifically when you travel using a vehicle or on wheels:

fahren – go/drive/ride using something with wheels or an engine (car, bus, bike, scooter, etc.).
gehen – go on foot (walk).

Since mit dem Roller (with the scooter) clearly means you’re using a vehicle, fahren is the natural choice:

Ich fahre mit dem Roller … = I drive/ride the scooter to …

If you were walking, you would say:

Ich gehe nach der Arbeit zum Spielplatz … = I walk to the playground …

Why is it nach der Arbeit and not nach die Arbeit or just nach Arbeit?

Several things are happening here:

Preposition + case
The preposition nach (meaning after in a temporal sense) always takes the dative case.
Gender and dative form
Arbeit is feminine:
- Nominative: die Arbeit
- Dative singular: der Arbeit
So with nach you must say: nach der Arbeit.
Why not nach Arbeit?
You could say nach Arbeit in some very abstract or technical contexts, but for everyday “after work” you almost always use the article:
- Nach der Arbeit fahre ich nach Hause. = After work I go home.

So nach der Arbeit is the standard, idiomatic way to say after work.

Why is it mit dem Roller and not mit den Roller or mit der Roller?

Again, this is about case and gender:

The preposition mit always takes the dative case.
Roller (scooter) is masculine:
- Nominative: der Roller
- Dative singular: dem Roller

So:

mit + dem Roller → mit dem Roller

mit den Roller would be wrong for singular.
mit den Rollern could be correct for plural (Roller → Rollern in dative plural):

mit den Rollern = with the scooters.

What exactly is a Roller in German? Is it a motorbike, a scooter, or something else?

Roller in German generally means a scooter. Depending on context, it can be:

A motor scooter (small motorcycle with a step‑through frame).
A kick scooter (the thing you push along with one foot).

If people mean a bicycle, they say Fahrrad, and for a motorcycle they say Motorrad.

So mit dem Roller = by/with the scooter, not a regular bike. Context or extra words (e.g. Motorroller) can make it more specific.

Why is it zum Spielplatz instead of zu dem Spielplatz or in den Spielplatz?

zu dem → zum
- zu
  - dem contract to zum in standard German.
- So zu dem Spielplatz and zum Spielplatz mean the same, but zum is more natural.
Why zu here?
zu is commonly used for going to a place (often a person, building, or some destination):
- zum Arzt (to the doctor)
- zur Schule (to school)
- zum Spielplatz (to the playground)
Why not in den Spielplatz?
- in den would be used if you stress going into an enclosed space (like a room or building).
- A Spielplatz is usually an open area, so zum Spielplatz (to the playground) or auf den Spielplatz (onto the playground) is more idiomatic.

You could also hear:

Ich fahre mit dem Roller auf den Spielplatz.
This focuses slightly more on ending up on the grounds of the playground. Both zum and auf den are common.

Why is Spielplatz masculine (so we get zum Spielplatz) and not feminine or neuter?

In German, the grammatical gender of a noun is mostly arbitrary and must be learned:

Spielplatz = the playground
Article: der Spielplatz (masculine)

So:

Nominative: der Spielplatz
Dative singular: dem Spielplatz

With zu (which takes dative), you get:

zu dem Spielplatz → zum Spielplatz

There is no logical rule that forces Spielplatz to be masculine; it’s simply how the word is defined in German.

What is the function of um … zu sehen here, and why do we need um and zu?

um … zu + infinitive expresses purpose (the reason why you do something).

Structure:

um
- [rest of infinitive clause] + zu
  - [infinitive verb]

In the sentence:

um das Baby kurz zu sehen
- um = introduces the purpose
- das Baby kurz = object + adverb inside the clause
- zu sehen = infinitive of sehen with zu

Whole meaning: in order to see the baby briefly.

We use um … zu when:

The subject of the main clause and the infinitive clause is the same (here: ich in both).

If the subject were different, you would typically use damit instead, e.g.:

Ich fahre zum Spielplatz, damit meine Freundin das Baby sehen kann.
(… so that my girlfriend can see the baby.)

Why is there a comma before um? Is it mandatory?

Yes, in this case it is mandatory according to standard German punctuation rules.

An infinitive clause with um, ohne, statt/anstatt, außer, als must be separated by a comma.

So:

…, um das Baby kurz zu sehen.

The comma marks the beginning of the infinitive purpose clause.
Leaving out this comma is considered a spelling/punctuation error in standard written German.

Why is it das Baby and not der Baby or die Baby? Does German always use neuter for “baby”?

Baby in German is grammatically neuter:

das Baby (nominative/accusative singular)

This is true regardless of the actual biological sex of the child.

Examples:

Das Baby schläft. (The baby is sleeping.)
Ich sehe das Baby. (I see the baby.)

You might still refer to your baby using a gendered word:

mein Sohn (my son)
meine Tochter (my daughter)

But the noun Baby itself remains neuter: das Baby.

What does kurz mean here? Literally “short”, or does it mean “briefly/for a moment”?

In this context, kurz is an adverb and means briefly / for a short while / for a moment, not “physically short”.

So:

…, um das Baby kurz zu sehen.
= “… in order to see the baby briefly / to see the baby for a moment.”

Compare:

ein kurzes Treffen – a short meeting (adjective)
Wir treffen uns kurz. – We’ll meet briefly (adverb)

So here it’s about the duration of seeing the baby.

Is the word order inside um das Baby kurz zu sehen fixed? Could I also say um kurz das Baby zu sehen?

Inside the um … zu clause, the finite verb is at the end as an infinitive, but the order of objects and adverbs is somewhat flexible.

Possible variants:

um das Baby kurz zu sehen (given sentence)
um kurz das Baby zu sehen

Both are grammatically correct.

Nuances:

um das Baby kurz zu sehen – slightly more neutral; puts the focus first on what you see (the baby), then how long (kurz).
um kurz das Baby zu sehen – can sound a bit more like only briefly is being emphasized.

In everyday speech, many speakers would use both without much distinction. The original word order is very natural.

Why is the present tense (fahre) used here instead of a future form like werde fahren?

German often uses the present tense to talk about the near future, especially when the time reference is clear from context:

Morgen fahre ich nach Berlin. – I’m going to Berlin tomorrow.
Heute Abend gehe ich ins Kino. – I’m going to the cinema this evening.

In your sentence, Manchmal fahre ich … describes a habitual action (something you sometimes do), not a single future plan, so the present tense is exactly right.

You could use werde fahren for a more specific, one‑time future intention, but that would change the meaning to I will (on this specific occasion) go by scooter … instead of a general habit.