Jeg tager min rygsæk med i metroen, men jeg lægger pungen i min lomme.

Because tage med is a fixed expression meaning take along / bring with you.

So:

• tage = take
• tage ... med = take something along

In this sentence, med is not just an extra word meaning with. It is part of the verb phrase.

Examples:

• Jeg tager en bog med. = I’m taking a book along.
• Hun tager sin jakke med. = She brings her jacket.

In Danish, small verb particles like med often come after the object in a main clause.

So the natural order is:

• Jeg tager min rygsæk med.

not usually:

• Jeg tager med min rygsæk.

That second version would sound like I go/take using my backpack or otherwise be interpreted differently. In your sentence, med belongs to tage med, so it comes after the noun phrase min rygsæk.

Lægger is the present tense of lægge, which means to lay / put something somewhere.

Danish often distinguishes between:

• lægge = put something down, lay it somewhere
• stille = place something standing upright
• sætte = set something down, often onto a seat/surface or into a position
• putte = put, tuck, insert

A wallet is naturally something you lægger or putter in a pocket. So:

• jeg lægger pungen i min lomme = I put the wallet in my pocket

A useful related pair is:

• lægge = to lay/put
• ligge = to lie/be lying

So lægge is the action, and ligge is the resulting position.

After a possessive like min, Danish normally uses the indefinite form of the noun.

So:

• min rygsæk = my backpack
• min lomme = my pocket

Not:

• min rygsækken
• min lommen

This is similar to English: you say my backpack, not my the backpack.

Because Danish often uses the definite form when the thing is already understood or easily identifiable from the context.

So pungen means the wallet, but in context it can still clearly mean my wallet.

Here, both can work:

• jeg lægger pungen i min lomme
• jeg lægger min pung i min lomme

The version with pungen sounds a bit more neutral and natural in many contexts.
The version with min pung puts slightly more focus on ownership.

English often uses possessives where Danish is happy with the definite form.

Metroen is the definite singular form of en metro.

• en metro = a metro
• metroen = the metro

In Danish, transport and locations are often referred to in the definite form when they are understood as the relevant or usual one in the situation.

So:

• i metroen = in/on the metro

If you said i en metro, it would mean in a metro train / in a metro, with a more indefinite, nonspecific meaning.

I usually means in / inside.

That fits both parts of the sentence:

• i metroen = inside the metro
• i min lomme = inside my pocket

For lomme, i is the obvious choice because a pocket is something you put things inside.

For transport, Danish often uses i when the idea is being physically inside the vehicle. So i metroen is very natural here.

They are both in the present tense.

• tage → tager
• lægge → lægger

In Danish, the present tense is often formed by adding -r to the infinitive.

The present tense can describe:

• something happening now
• a habitual action
• a general fact
• sometimes even near future actions

So this sentence could describe a routine or a typical action, not only something happening at this exact second.

Yes. Danish very often writes compound nouns as one word.

So:

• ryg = back
• sæk = sack/bag
• rygsæk = backpack

This is extremely common in Danish. English sometimes writes similar ideas as two words, but Danish usually joins them into one compound noun.

Because men joins two full clauses, and each clause needs its own subject.

So the structure is:

• Jeg tager min rygsæk med i metroen
• men jeg lægger pungen i min lomme

Both halves are complete clauses, so repeating jeg is necessary.

That works the same way as in English:

• I take my backpack on the metro, but I put the wallet in my pocket.