Nach dem Regen ist der Weg voller Schlamm.

Here nach means after, so nach dem Regen means after the rain.

German nach can also mean other things in other contexts, such as:

• to/toward: nach Berlin
• according to: nach dem Plan

But in this sentence, it is a time expression: after the rain.

Because nach takes the dative case here.

The basic form is der Regen (the rain).
In the dative singular, that becomes dem Regen.

So:

• der Regen = nominative
• dem Regen = dative

That is why German says nach dem Regen, not nach der Regen.

Because German main clauses usually follow the verb-second rule.

The first position in the sentence is taken by Nach dem Regen.
So the finite verb must come next:

• Nach dem Regen | ist | der Weg voller Schlamm.

If you put the subject first, that is also correct:

• Der Weg ist nach dem Regen voller Schlamm.

Both are grammatical. The original version emphasizes the time setting a bit more.

Yes, absolutely.

That version starts with the subject der Weg, while the original starts with the time phrase nach dem Regen.

So both work:

• Nach dem Regen ist der Weg voller Schlamm.
• Der Weg ist nach dem Regen voller Schlamm.

The difference is mainly one of focus and style, not basic meaning.

Weg is a general word for a path, way, track, or sometimes road.

In this sentence, it most naturally suggests a path or track, especially because mud fits that kind of setting well.

So depending on context, English might translate Weg as:

• path
• track
• way
• sometimes road

Voll means full.

When German says what something is full of, a very common pattern is voller + noun:

• voller Schlamm = full of mud
• voller Wasser = full of water
• voller Leute = full of people

So:

• Der Weg ist voll. = The path is full.
• Der Weg ist voller Schlamm. = The path is full of mud.

A good practical way to learn this is to treat voller + noun as a common full of ... pattern.

Because Schlamm is usually treated as an uncountable substance noun, like mud in English.

When you mean the substance in a general sense, German often uses no article:

• voller Schlamm = full of mud

That is similar to English, where you would normally say full of mud, not full of the mud.

Yes.

Schlammig means muddy, so:

• Der Weg ist schlammig. = The path is muddy.

This is a bit simpler and more general.
Voller Schlamm sounds a little more vivid and suggests there is a lot of mud there.

So the difference is roughly:

• schlammig = muddy
• voller Schlamm = full of mud / covered with a lot of mud

Because they are nouns.

In German, all nouns are capitalized:

• der Regen
• der Weg
• der Schlamm

Nach is also capitalized here, but only because it is the first word of the sentence, not because it is a noun.