Después de la lluvia, vemos una huella de caballo en el barro.

Because después needs the preposition de before a noun.

• después de la lluvia = after the rain
• después de comer = after eating
• después de eso = after that

So después on its own is not enough here. De is required.

Spanish often uses the definite article more than English does.

So:

• la lluvia = the rain
• el barro = the mud
• la nieve = the snow

Even when English would sound more natural without the, Spanish often keeps the article. In this sentence, después de la lluvia is the normal way to say after the rain.

Because Después de la lluvia is an introductory time phrase.

The comma helps separate that opening phrase from the main clause:

• Después de la lluvia, vemos...

In Spanish, this comma is very common and natural. It is similar to English:

• After the rain, we see...

Sometimes a short introductory phrase may appear without a comma, but here the comma is perfectly standard.

Spanish usually drops subject pronouns when they are not needed.

The ending -emos already tells you the subject is we:

• veo = I see
• ves = you see
• ve = he/she sees
• vemos = we see

So vemos already means we see. Adding nosotros is possible, but it is usually only used for emphasis, contrast, or clarity.

• Nosotros vemos una huella, pero ellos no.
  = We see a footprint, but they don’t.

In a neutral sentence, vemos is enough.

Because Spanish often uses the present tense to describe what is happening in a scene, what we notice, or what the sentence presents as a current observation.

• vemos una huella... = we see a footprint...

This can sound like:

• a general statement,
• a description of a scene,
• or something happening now.

It works just like English present tense in sentences such as We see a footprint in the mud.

Because huella is a feminine noun:

• la huella
• una huella

The h is silent, so pronunciation begins with the ue sound, but that does not change its gender. It is simply a feminine noun.

Also, huella usually means a mark, track, print, or footprint, depending on context.

They are similar, but not always identical.

• huella = a mark, track, footprint, trace left behind
• pisada = a step or footprint, often focusing more directly on the act of stepping

In this sentence, huella is a very natural choice because it refers to the visible mark left in the mud.

A huella de caballo is basically a horse’s print/track in the mud. Depending on context, English might translate it as:

• footprint
• hoofprint
• track

Because de caballo here describes the type or origin of the mark, not a specific horse.

• una huella de caballo = a horse track / a horse’s track / a hoofprint
• una huella del caballo = the track of the horse

So:

• de caballo = from a horse / horse-type / horse-related, non-specific
• del caballo = of the horse, a specific horse already known in context

If the sentence is introducing the track for the first time, de caballo is the natural choice.

Because Spanish often leaves out the article after de when the noun is being used in a general, descriptive way.

• huella de caballo = horse track / hoofprint
• casa de piedra = stone house
• zapatos de cuero = leather shoes

If you say de un caballo, you are stressing that it belongs to one horse in particular, but still not identifying which one:

• una huella de un caballo = a track from a horse

That is grammatically possible, but less natural here. Huella de caballo sounds more like the normal label for the kind of print it is.

Because en el barro suggests the print is in the mud, impressed into it, which is exactly what a hoofprint or footprint does.

• en el barro = in the mud
• sobre el barro = on top of the mud

A huella is normally something left in a soft surface, so en is the better choice.

Yes, Spanish word order is flexible, although some versions sound more natural than others.

The original sentence:

• Después de la lluvia, vemos una huella de caballo en el barro.

Other possible orders include:

• Vemos una huella de caballo en el barro después de la lluvia.
• En el barro, vemos una huella de caballo después de la lluvia.

But the original order is very natural because it starts with the time phrase and then gives the main observation.

Spanish often puts time or place expressions first when setting the scene.

Usually no, not with the same meaning.

• ver = to see
• mirar = to look at

So:

• vemos una huella = we see a track
• miramos una huella = we look at a track

In your sentence, the idea is that the track is visible to us, so vemos is the right verb. If the meaning were that we are deliberately looking at it, then miramos could work.

Usually, yes, in this kind of sentence.

Barro commonly means:

• mud
• clay
• wet earth

Here, because of the rain and the track, mud is the best translation.

Yes, it is grammatical, but it changes the feel slightly.

• una huella de caballo = a horse track / hoofprint
• una huella de un caballo = a track from a horse

The version without un sounds more general and more idiomatic here. The version with un feels a little more specific or analytical, as if you are stressing that the track came from one horse.

So in ordinary Spanish, una huella de caballo is the more natural choice in this sentence.