Nach dem Picknick trage ich den Müll weg.

Because nach takes the dative case here.

• Base form: das Picknick
• Dative form: dem Picknick

So:

• nach dem Picknick = after the picnic

This is a very common pattern:

• nach dem Essen = after the meal
• nach der Arbeit = after work
• nach dem Film = after the film

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

In this sentence, Nach dem Picknick comes first, so the verb must come next:

• Nach dem Picknick | trage | ich den Müll weg.

Even though ich is the subject, it comes after the verb because the first position is already taken by the time phrase.

The full verb is wegtragen.

It is a separable verb:

• weg- = away
• tragen = to carry

In a main clause, the prefix separates:

• Ich trage den Müll weg.

In the infinitive, it stays together:

• Ich will den Müll wegtragen.

So trage is the conjugated part, and weg goes to the end.

Because wegtragen is a separable verb.

In a normal present-tense main clause:

• the conjugated verb goes in second position
• the separable prefix goes to the end

So:

• Ich trage den Müll weg.

This is similar to:

• Ich stehe früh auf. from aufstehen
• Sie räumt das Zimmer auf. from aufräumen

Because den Müll is the direct object, so it is in the accusative case.

The noun is:

• nominative: der Müll
• accusative: den Müll

Here, ich is doing the action, and den Müll is what is being carried away.

So:

• ich = subject
• den Müll = direct object

Müll usually means trash/garbage in a general sense. It is often treated like a mass noun, similar to English trash or garbage.

So den Müll here means:

• the trash
• the rubbish
• the garbage

It does not usually mean one single item.

Yes, tragen by itself means to carry or to wear, depending on context. In this sentence, with weg-, it means to carry away.

So den Müll wegtragen is literally:

• to carry the trash away

In natural English, this may be translated as:

• take away the trash
• carry out the trash
• remove the trash

The exact English wording depends on context.

Yes. That is also grammatical.

Possible word orders include:

• Nach dem Picknick trage ich den Müll weg.
• Ich trage den Müll nach dem Picknick weg.
• Ich trage nach dem Picknick den Müll weg.

They all mean roughly the same thing, but the emphasis changes a little.

• Nach dem Picknick ... puts the time phrase in focus first.
• Ich trage ... starts with the subject, which is a more neutral English-like order.

No, not in standard German.

If Nach dem Picknick is placed first, the verb must still stay in the second position:

• correct: Nach dem Picknick trage ich den Müll weg.
• incorrect: Nach dem Picknick ich trage den Müll weg.

This is one of the most important German word-order rules.

Because all German nouns are capitalized.

So:

• das Picknick
• der Müll

This is true even in the middle of a sentence.

Here, nach is a preposition followed by a noun phrase:

• nach dem Picknick = after the picnic

Nachdem is a conjunction that introduces a subordinate clause:

• Nachdem wir gepicknickt haben, trage ich den Müll weg.
• After we had a picnic, I carry away the trash.

So the difference is:

• nach
  • noun
• nachdem
  • full clause

It is possible and understandable, especially if you mean physically carrying it away. But depending on context, Germans might also use other verbs, such as:

• den Müll wegbringen = take the trash away
• den Müll rausbringen = take the trash out
• den Müll entsorgen = dispose of the trash

So wegtragen is fine, but it sounds especially physical: you are literally carrying it away.