Mi abuela me enseñó a desatar los cordones despacio para no romperlos.

Me is the indirect object pronoun meaning to me.

So:

• Mi abuela enseñó... = My grandmother taught...
• Mi abuela me enseñó... = My grandmother taught me...

It marks the person who received the teaching. In this sentence, your grandmother is the one doing the teaching, and me is the person learning.

Enseñó is the preterite form of enseñar, third person singular:

• yo enseñé
• tú enseñaste
• él/ella enseñó

Here it means she taught as a completed action in the past.

Why not the others?

• enseña = she teaches / is teaching
• enseñaba = she used to teach / was teaching

So me enseñó is the natural choice if you mean one completed act of teaching, or you are simply narrating a past event.

After enseñar, Spanish often uses a + infinitive to mean to teach someone to do something.

Pattern:

• enseñar a + infinitive = to teach to do something

Examples:

• Me enseñó a cocinar = She taught me to cook
• Me enseñó a nadar = She taught me to swim
• Me enseñó a desatar los cordones = She taught me to untie the laces

So the a is required by this structure.

The understood subject is me — the person being taught.

So the sentence means your grandmother taught you:

• to untie the laces slowly
• so as not to break them

Even though Spanish infinitives do not show the subject directly, it is normally understood from the context. With enseñar a + infinitive, the action of the infinitive is usually done by the person being taught.

Spanish often uses the definite article where English might use no article at all.

So:

• desatar los cordones = to untie the shoelaces / laces

Even if English often says just untie shoelaces, Spanish usually prefers los cordones.

This is very normal with body parts, clothing, and familiar objects:

• Me lavé las manos = I washed my hands
• Se quitó la chaqueta = He took off his jacket
• Desató los cordones = He untied the laces

Yes, in this context cordones means shoelaces.

By itself, cordón can mean a cord, lace, or string, but with desatar and everyday context, los cordones is commonly understood as shoelaces.

A useful Spain-related note: cordones is a safe, standard word here. In some other varieties of Spanish, learners may encounter other words for shoelaces, but in Spain cordones is very natural.

Despacio means slowly, and here it most naturally modifies desatar.

So the idea is:

• to untie the laces slowly

In other words, your grandmother taught you to do the untying carefully and slowly.

The placement is quite natural:

• Me enseñó a desatar los cordones despacio

Spanish adverbs are fairly flexible, but placement can slightly affect emphasis. In this sentence, despacio sounds closely linked to the action of untying.

Yes, but the nuance can shift a little.

For example:

• Mi abuela me enseñó a desatar los cordones despacio
  Most naturally: she taught me to untie them slowly

• Mi abuela me enseñó despacio a desatar los cordones
  This can suggest she taught me slowly, meaning the teaching itself was slow

So the original word order is good if the point is that the laces should be untied slowly.

This is a very common structure:

• para + infinitive = in order to
• para no + infinitive = in order not to

So:

• para no romperlos = in order not to break them

It expresses purpose. The idea is that the laces should be untied slowly so that they do not get broken.

This is a very useful pattern:

• Estudio para aprender = I study in order to learn
• Habla bajo para no despertarlo = Speak quietly so as not to wake him
• Desátalos despacio para no romperlos = Untie them slowly so as not to break them

Los refers back to los cordones.

Because cordones is:

• masculine
• plural

the pronoun must also be masculine plural:

• los

So:

• romperlos = to break them
• them = the laces

If the noun were feminine plural, you would use las instead.

Because Spanish allows object pronouns to attach to an infinitive.

So:

• romperlos = to break them

This is completely normal after para no:

• para no romperlos

Spanish object pronouns can be attached to:

• infinitives: hacerlo
• gerunds: haciéndolo
• affirmative commands: hazlo

In this sentence, attaching los to romper is the standard option.

No, that would not be correct here.

With a bare infinitive like romper, the object pronoun goes attached:

• para no romperlos ✔
• para no los romper ✘

You can place pronouns before a conjugated verb:

• no los rompas = don’t break them

But with an infinitive by itself, the pronoun attaches to the end.

Because the sentence means to break them, not to break oneself or to get broken in a reflexive/passive-like way.

• romperlos = to break them
• romperse = to break oneself / to break / to get broken, depending on context

Here the focus is on avoiding damage to the laces, so the direct object pronoun los is needed.

In everyday speech, despacio sounds very natural.

Both can mean slowly, but:

• despacio = common, everyday, very natural in speech
• lentamente = also correct, but can sound a bit more formal or deliberate depending on context

So in this sentence, despacio is an excellent choice. It sounds like natural spoken Spanish.