Prima di entrare, pulisci bene gli stivali sullo zerbino, per favore.

Literally, prima di entrare means before entering.

Italian often uses prima di + infinitive when the person doing both actions is the same. Here, the same person is being told to do both things:

• entrare = to enter
• pulisci = clean

So the structure is very natural: before entering, clean...

If Italian wanted to mention a different subject, it would usually use a different structure, such as prima che + subjunctive.

Here, pulisci is a command.

It is the tu imperative form of pulire. For many regular -ire verbs, the tu imperative looks exactly like the present tense:

• tu pulisci = you clean
• pulisci! = clean!

So only the context tells you whether it is a statement or a command. In this sentence, the punctuation and tone make it clearly a command.

It is informal singular, addressed to one person as tu.

That is why the verb is pulisci.

Other versions would be:

• informal singular: pulisci
• formal singular: pulisca
• plural: pulite

So you could say:

• Prima di entrare, pulisca bene gli stivali sullo zerbino, per favore.
  = formal, to one person

• Prima di entrare, pulite bene gli stivali sullo zerbino, per favore.
  = to more than one person

Because stivali begins with s + consonant: st-.

In Italian, masculine plural nouns that would take lo in the singular take gli in the plural:

• lo stivale
• gli stivali

So i stivali would be wrong here.

This is the same pattern as:

• lo studente / gli studenti
• lo specchio / gli specchi

Italian often uses the definite article where English uses a possessive, especially with clothing, body parts, and personal belongings when ownership is obvious from the context.

So:

• pulisci bene gli stivali

naturally means:

• clean your boots well

Even though the sentence literally says the boots, Italian speakers understand whose boots they are.

If you said i tuoi stivali or, more correctly here, gli stivali tuoi / i tuoi stivali? Actually with stivali, the normal form is i tuoi stivali only if the noun takes i, but since it does not, the correct form is i tuoi stivali? No — the article with a possessive changes: with possessives, you use the normal plural article for masculine plural nouns, which is i. So i tuoi stivali is correct.
That would add emphasis, as in your boots specifically.

Bene means well.

In this sentence, it suggests properly or thoroughly:

• pulisci bene gli stivali = clean your boots well / clean your boots properly

Italian often puts adverbs like bene after the verb, so this word order is very normal.

Sullo is a contraction of:

• su = on
• lo = the

So:

• su + lo = sullo

You need lo because zerbino is a masculine singular noun beginning with z, and such nouns take lo:

• lo zerbino
• sullo zerbino = on the doormat

Compare:

• il tavolo → sul tavolo
• lo zerbino → sullo zerbino

Yes, exactly.

Here sullo zerbino means on the doormat. The idea is that you wipe or clean the boots using the surface of the mat.

That is why su makes sense: the action happens on the doormat.

Using something like nello zerbino would sound wrong, because that would suggest being in the doormat rather than on it.

Zerbino means doormat.

It is the mat by the door where people wipe their shoes or boots before coming in.

A useful note: zerbino can also be used figuratively in Italian to describe a person who lets others walk all over them, like English doormat. But in this sentence it is completely literal.

Per favore means please.

It is often placed at the end of a request in Italian, especially in everyday speech:

• Pulisci bene gli stivali, per favore.

That said, it can also appear in other positions:

• Per favore, pulisci bene gli stivali.
• Pulisci, per favore, bene gli stivali.

The last one is possible, but less natural. The most natural positions are usually the beginning or the end.

Putting per favore at the end sounds polite and very normal.

Yes. Italian word order is fairly flexible.

For example, you could also say:

• Pulisci bene gli stivali sullo zerbino prima di entrare, per favore.

That still means the same thing.

Starting with Prima di entrare gives a slight emphasis to the time sequence: before you come in, do this first. That is why the original version sounds especially natural for an instruction at a doorway.