Después de la playa, quiero quitarme el bañador y ducharme.

Because después de is a fixed expression in Spanish. When después is followed by a noun, pronoun, or infinitive, it normally takes de.

Examples:

• después de clase
• después de eso
• después de comer

So después de la playa is the correct structure.

Yes, literally it is after the beach, but in natural English it usually means after going to the beach or after being at the beach.

Spanish often uses this kind of shorthand. If you wanted to be more explicit, you could say después de ir a la playa, but después de la playa is very natural.

Because querer is followed directly by an infinitive in Spanish.

So you say:

• quiero comer
• quiero salir
• quiero ducharme

English uses want to, but Spanish does not add an extra word there.

Because querer is a stem-changing verb. In many forms, the e changes to ie.

So:

• quiero
• quieres
• quiere
• queremos
• queréis
• quieren

That is why the sentence uses quiero.

The me is a reflexive pronoun. It shows that the action is being done to yourself.

• quitarme = to take off from myself
• ducharme = to shower myself

In English we usually just say take off my swimsuit and shower, but Spanish often marks this more explicitly.

Quitar means to remove or to take away something.

Quitarse means to take something off oneself.

So:

• quitar el bañador = remove the swimsuit
• quitarse el bañador = take off your own swimsuit

In this sentence, the reflexive form is needed because the person is removing their own swimsuit.

Spanish often uses the definite article with clothing, body parts, and personal items when it is already clear whose they are.

Here, me already tells you it is my swimsuit, so el bañador sounds natural.

So Spanish prefers:

• quitarme el bañador

rather than:

• quitarme mi bañador

The second one is not impossible, but it is less natural in this context.

It is especially common in Spain, which fits your sentence.

In Spain, bañador is a very normal word for swimsuit or swimming trunks. In many parts of Latin America, people are more likely to say traje de baño.

So this sentence sounds very natural in Spanish from Spain.

Because both infinitives depend on the same verb quiero.

The structure is:

• quiero
  • quitarme el bañador
• y
  • ducharme

This is just like English:

• I want to take off my swimsuit and shower

You do not need to repeat quiero unless you want extra emphasis.

Yes. With a conjugated verb plus an infinitive, the pronoun can often go in either place:

• quiero quitarme el bañador
• me quiero quitar el bañador

Both are correct. In your sentence, quiero quitarme el bañador y ducharme sounds very natural and smooth.

For talking about showering yourself, ducharse is the normal form.

So:

• quiero ducharme = I want to take a shower / shower myself

Without the reflexive, duchar usually means showering someone else.

For example:

• Voy a duchar al niño = I’m going to shower the child

So in your sentence, ducharme is exactly the expected form.