Antes de dormir, me pongo la pomada y luego me cambio la venda con cuidado.

In Spanish, antes de must be followed by a noun or an infinitive.

• antes de dormir = before sleeping / before going to sleep
• antes de la cena = before dinner

So de is required here.
Antes dormir is not correct standard Spanish.

Because antes de + infinitive often uses the most basic verb form when you mean the action in a general way.

• antes de dormir = before sleeping / before going to sleep

You could also hear:

• antes de dormirme = before falling asleep
• antes de irme a dormir = before going to bed / before going to sleep

Those are possible, but they change the nuance a bit.
In your sentence, antes de dormir is the most neutral and natural choice.

Here, ponerse means to put/apply something on oneself.

So:

• pongo la pomada would sound incomplete or would usually mean I put the ointment down / I place the ointment
• me pongo la pomada means I put the ointment on myself

That me shows that the action is done to your own body.

This is very common in Spanish with actions involving the body or personal care:

• me lavo las manos = I wash my hands
• me cepillo los dientes = I brush my teeth
• me pongo crema = I put cream on

Yes, in practical learner terms, it is treated as a reflexive or pronominal structure: ponerse algo.

It means to apply/put something on oneself.

Grammatically, you can think of it like this:

• me = on myself / to myself
• la pomada = the thing being applied

So the sentence is not saying the ointment itself is reflexive. It is saying I apply the ointment to myself.

Spanish often uses the definite article with body-related and personal-care contexts where English would use a possessive.

So Spanish prefers:

• me pongo la pomada
• me cambio la venda
• me lavo las manos

Instead of:

• me pongo mi pomada
• me cambio mi venda
• me lavo mis manos

Using mi is not impossible, but it usually sounds unnecessary unless you want to emphasize ownership, contrast, or clarity.

Because cambiarse algo can mean to change something on oneself or to replace something one is wearing/using on one’s own body.

So:

• me cambio la venda = I change my bandage / I replace the bandage on myself
• cambio la venda = I change the bandage

The second one is possible, but it is less personal and could refer to changing a bandage in general, or changing someone else’s bandage, depending on context.
The me makes it clear that the speaker is doing this to their own body.

Because the sentence is simply talking about the bandage as the known, relevant one in the situation.

• me cambio la venda = I change the bandage

In context, this naturally means I replace the bandage I have on.

If you wanted to be more specific, you could say:

• me cambio la venda por una nueva = I change the bandage for a new one
• me pongo otra venda = I put on another bandage

But the original sentence sounds perfectly natural as it is.

Venda usually means bandage.

Depending on context, English might translate it as:

• bandage
• dressing
• wrap

If someone has an injury covered with fabric or gauze, venda is a very normal word.

Both are things you put on the skin, but they are not always the same.

• pomada usually means ointment
• crema usually means cream

Pomada often sounds thicker, more medicinal, or more ointment-like.
Crema is more like a cream or lotion.

So la pomada in your sentence is specifically the ointment.

Because Antes de dormir is an introductory time phrase at the beginning of the sentence.

So the comma helps separate:

• Antes de dormir, = before going to sleep,
• me pongo la pomada... = I put on the ointment...

In Spanish, this comma is very natural and standard. Sometimes commas in short introductory phrases can be omitted, but here the comma is a good, clear choice.

Luego here means then or afterwards.

So:

• me pongo la pomada y luego me cambio la venda = I put on the ointment and then I change the bandage

Yes, después would also work:

• ...y después me cambio la venda

Both are common.
Luego is very natural in Spain.

Yes, luego is flexible.

The original order is very natural:

• me pongo la pomada y luego me cambio la venda

You could also say:

• me pongo la pomada y me cambio luego la venda
• luego me cambio la venda

But the original version is the clearest and most natural for learners.
Placing luego before the second verb is a very common way to mark sequence.

Con cuidado means carefully or more literally with care.

So:

• me cambio la venda con cuidado = I change the bandage carefully

Spanish often uses a phrase with con + noun where English might prefer an adverb:

• con cuidado = carefully
• con calma = calmly
• con paciencia = patiently

Yes, but con cuidado is much more common and natural in everyday speech.

• me cambio la venda con cuidado = very natural
• me cambio la venda cuidadosamente = correct, but a bit more formal or less conversational

For normal daily Spanish, con cuidado is usually the better choice.

Because the present tense in Spanish is often used for habitual actions and routines.

So:

• Antes de dormir, me pongo la pomada...
  means something like
  Before going to sleep, I put on the ointment...

It describes what the speaker normally does.

This is the same idea as:

• Por la mañana me ducho = In the morning I shower
• Después de cenar, me lavo los dientes = After dinner, I brush my teeth

Yes. The Spanish present tense can express:

• a habitual action
• a current action
• a near-future action, depending on context

So without more context, this sentence could mean:

• This is what I do before bed
  or
• Right now, before sleeping, I’m putting on the ointment and then changing the bandage

Very often, though, a sentence like this sounds like a routine unless the context says otherwise.

Because there are two separate actions done to the speaker’s own body:

• me pongo la pomada = I apply the ointment to myself
• me cambio la venda = I change the bandage on myself

Each verb needs its own pronoun because each action is personal to the speaker.

Spanish does this very often in self-care actions:

• me lavo la cara y me cepillo los dientes
• me quito la chaqueta y me pongo el pijama

Yes. Me aplico la pomada is also correct and often sounds a bit more precise or slightly more formal/medical.

• me pongo la pomada = very common, everyday
• me aplico la pomada = also correct, a bit more technical or explicit

Both are natural, but me pongo la pomada is perfectly normal in ordinary speech.