Po očkování hladím štěně v pelíšku a starám se o něj.

Because po meaning after normally takes the locative case in Czech.

• očkování = vaccination / inoculation
• after po, it becomes po očkování

In this noun, the locative form looks the same as the basic form, so there is no visible change.

It is a noun here, not a finite verb.

More specifically, it is a verbal noun formed from the verb očkovat (to vaccinate). Czech often uses these verbal nouns in places where English might also use a noun:

• po očkování = after the vaccination / after vaccinating

So the sentence does not mean someone is actively vaccinating at that moment; it refers to the event as a thing.

Hladím is the 1st person singular present tense form of hladit (to stroke, pet, caress).

So:

• hladit = infinitive
• hladím = I stroke / I am stroking / I pet

Czech present tense often covers both the simple present and the present continuous, depending on context.

Because Czech usually does not need subject pronouns when the verb ending already shows who the subject is.

Here, hladím clearly means I stroke / I am stroking, and starám se also matches I take care.

You could say Já po očkování hladím..., but já would usually add emphasis, such as I do it.

Because štěně is a neuter noun, and in this sentence it is the direct object, so it is in the accusative case.

For many neuter nouns, especially this type, the nominative and accusative forms are identical:

• nominative: štěně = puppy
• accusative: štěně = puppy as a direct object

So the form does not change, even though the case does.

Pelíšek means something like pet bed, little bed, or basket—a cozy place where an animal sleeps.

After the preposition v meaning in, Czech uses the locative case when talking about location:

• basic form: pelíšek
• locative after v: v pelíšku

Also, pelíšek is a diminutive, which gives it a cozy or affectionate feeling.

Because the verb is starat se o..., which means to take care of...

The se is part of the verb itself. In other words, the full verb is not just starat, but starat se.

So:

• starám se = I take care
• starám se o něj = I take care of him/it

Many Czech verbs include se or si, and learners usually need to memorize them as part of the whole verb.

Because starat se is a verb that is normally followed by o + accusative.

So the pattern is:

• starat se o něco / někoho = to take care of something / someone

In this sentence:

• o něj = of him / it

This is just the standard construction used with this verb.

After a preposition, Czech personal pronouns usually take a special form beginning with n-.

So:

• without a preposition: ho
• after a preposition: něj

That is why Czech says:

• vidím ho = I see him/it
• starám se o něj = I take care of him/it

This n- form is very common after prepositions.

Not necessarily. Here něj refers back to štěně.

The noun štěně is grammatically neuter, but Czech uses the same oblique forms of the 3rd person pronoun for masculine and neuter in many cases, especially after prepositions:

• o něj
• k němu
• s ním

So this is completely normal for štěně.

Czech present tense is broader than English present tense in usage.

Hladím and starám se can mean:

• I pet
• I am petting
• I take care of
• I am taking care of

The exact English translation depends on context. Czech often leaves that distinction to the situation rather than changing the verb form.

Yes. Czech word order is fairly flexible because the case endings show grammatical roles.

This sentence:

• Po očkování hladím štěně v pelíšku a starám se o něj.

could be rearranged in some contexts, for example to emphasize a different part. But the original order sounds natural and neutral.

A rough information structure is:

• Po očkování = setting the time/context
• hladím štěně v pelíšku = main action
• a starám se o něj = added related action

So while other orders are possible, this one is a very normal choice.

Because hladit is imperfective, and that fits an ongoing, repeated, or general action.

In this sentence, the speaker describes what they are doing after the vaccination: petting the puppy and taking care of it. That sounds like a process or activity, not a single completed action.

If you used a perfective verb such as pohladím, it would usually suggest a single completed act of stroking, more like I will stroke / I give it a stroke, depending on context.

So hladím is the natural choice here.