Zum einen lerne ich neue Wörter mit Karten, zum anderen schreibe ich jeden Abend kurze Sätze.

It is a fixed expression meaning something like:

• for one thing ..., for another ...
• on the one hand ..., on the other hand ...

In this sentence, it introduces two different methods or aspects of learning:

• Zum einen = one method / one aspect
• zum anderen = another method / another aspect

It does not always sound as strongly contrasting as English on the one hand ... on the other hand .... Often it simply organizes two points clearly.

German main clauses follow the verb-second rule. That means the conjugated verb must come in the second position.

So in:

• Zum einen lerne ich neue Wörter ...

the phrase Zum einen takes the first position, so the conjugated verb lerne must come next. Then the subject ich comes after the verb.

The same happens in:

• Zum anderen schreibe ich ...

A useful pattern to remember:

• Today I learn ... in German structure becomes Heute lerne ich ...
• For one thing I learn ... becomes Zum einen lerne ich ...

Zum is the contraction of zu dem.

So literally, zum einen comes from zu dem einen. But in this expression, you should treat zum einen ..., zum anderen ... as a set phrase rather than translating each word separately.

So yes, zum is grammatically zu dem, but the whole expression is idiomatic.

Because these are two separate main clauses:

• Zum einen lerne ich neue Wörter mit Karten
• zum anderen schreibe ich jeden Abend kurze Sätze

German normally states the subject again in each full clause.

In English, we also often repeat it:

• For one thing, I learn new words with cards; for another, I write short sentences every evening.

You could sometimes shorten repeated material in speech, but in a normal sentence like this, repeating ich is the natural choice.

The comma separates two main clauses.

Each part has its own conjugated verb:

• lerne
• schreibe

So the structure is basically:

• main clause 1, main clause 2

German commonly uses a comma here to separate them. A semicolon would also be possible in some contexts, but the comma is very normal.

Mit always takes the dative case.

So mit Karten is dative plural.

Here, Karten already ends in -n in the plural, so you do not see any extra change. Compare:

• singular: mit der Karte
• plural: mit Karten

In context, Karten probably means something like cards or flashcards. A learner might also hear:

• mit Karteikarten
• mit Lernkarten

Those are more explicit ways to say flashcards.

Both Wörter and Sätze are plural nouns used without an article here.

When an adjective comes before a plural noun with no article, it takes the strong adjective ending. In the accusative plural, that ending is -e.

So:

• neue Wörter
• kurze Sätze

These are both direct objects of the verbs:

• ich lerne neue Wörter
• ich schreibe kurze Sätze

That is why the adjective endings look the same.

Because German often uses the accusative for time expressions without a preposition.

So:

• jeden Abend = every evening

This is a very common pattern:

• jeden Tag = every day
• jede Woche = every week
• jeden Morgen = every morning

Here, Abend is masculine, so jeder changes to jeden in the accusative.

German has two plural forms of Wort, and they are used differently:

• Wörter = individual words as vocabulary items
• Worte = words as part of a statement, saying, speech, or quotation

Since the sentence is about learning vocabulary, Wörter is the correct choice.

Examples:

• Ich lerne neue Wörter. = I am learning new vocabulary words.
• Seine Worte waren sehr freundlich. = His words were very kind.

Yes, often you can.

• einerseits ..., andererseits ...
• zum einen ..., zum anderen ...

Both can introduce two sides or two aspects.

However, einerseits ..., andererseits ... often feels a bit more like a contrast between two sides of an issue, while zum einen ..., zum anderen ... can sound a little more neutral and organized, especially when listing two points.

In this sentence, zum einen ..., zum anderen ... sounds very natural.

German often leaves out the article when speaking generally about plural countable things.

So:

• Ich lerne neue Wörter = I learn new words
• Ich schreibe kurze Sätze = I write short sentences

If you added an article, the meaning would become more specific:

• die neuen Wörter = the new words
• die kurzen Sätze = the short sentences

Without the article, the sentence sounds general, which fits the idea of a regular learning habit.

Mit Karten is understandable, especially if the context already makes it clear that these are study cards.

But many speakers would use a more specific word, such as:

• mit Karteikarten
• mit Lernkarten

So all of these are possible, with slightly different levels of precision:

• mit Karten = with cards
• mit Karteikarten = with index cards / flashcards
• mit Lernkarten = with study cards / flashcards

The sentence as given is still grammatically correct and easy to understand.