Learn Hardcore Portuguese (Portugal): Eu quero eliminar o desperdício de plástico em casa. - I want to eliminate plastic waste at home.

You don’t have to say Eu.
In Portuguese (especially European Portuguese), the subject pronoun is often dropped because the verb form quero already shows that the subject is eu.
So both are correct:

Eu quero eliminar o desperdício de plástico em casa.
Quero eliminar o desperdício de plástico em casa. (more natural in many contexts)

Why is eliminar in the infinitive after quero?

In Portuguese, when you use verbs like querer (to want), gostar de (to like), poder (can), precisar de (to need) followed by another verb, that second verb normally stays in the infinitive.
So Eu quero eliminar literally mirrors I want to eliminate:

quero = I want
eliminar = to eliminate (infinitive)
You don’t conjugate eliminar here: Eu quero elimino is incorrect.

Is Eu quero a bit too direct or rude in Portuguese? Should I say something softer?

Eu quero is grammatically fine and not rude by itself; it depends on context and tone.
However, for polite or formal situations, Portuguese speakers often soften it. For example:

Gostava de eliminar o desperdício de plástico em casa. – I’d like to eliminate…
Queria eliminar o desperdício de plástico em casa. – I wanted / I would like to eliminate…
These sound more tentative and polite, especially in European Portuguese.

What’s the difference between eliminar and acabar com here?

Both can work, but there’s a nuance:

eliminar o desperdício de plástico – sounds a bit more formal or “technical”, like eliminate plastic waste.
acabar com o desperdício de plástico – literally to put an end to plastic waste; this is very common and sounds a bit more conversational.
You could say:
Eu quero acabar com o desperdício de plástico em casa.

Why is it o desperdício and not just desperdício without the article?

Portuguese uses definite articles (o, a, os, as) more often than English uses the.
Here, o desperdício de plástico means the plastic waste / the wasting of plastic in general, not just any random instance.
You could say quero eliminar desperdício de plástico em casa, but it sounds less natural and more like talking about “some” waste; o desperdício de plástico sounds more complete and idiomatic.

What’s the difference between desperdício and lixo?

desperdício = waste in the sense of wastage, unnecessary use, something that shouldn’t have been used or thrown away. It focuses on the idea of wastefulness.
lixo = trash/garbage/rubbish, the physical stuff you throw away.
So:
desperdício de plástico = waste of plastic / plastic wastage
lixo de plástico = plastic trash (the actual discarded plastic objects)
Your sentence focuses on not wasting plastic, not just the trash itself.

Why is it de plástico and not do plástico?

de + o contracts to do, but only when you really mean of the.
Here, plástico is being used in a generic way, as a material: plastic in general, not the plastic.

desperdício de plástico = waste of plastic (as a material, in general)
desperdício do plástico would sound like waste of the plastic (some specific plastic already known in the context).
So de plástico is the normal, generic expression.

Could I say de plásticos (plural) instead of de plástico?

You could, but it changes the nuance:

desperdício de plástico = waste of plastic as a material, in general.
desperdício de plásticos = waste of plastic items / different types of plastics.
In most general sentences about environmental or household waste, de plástico (singular) is the standard, more natural option.

Can I change the word order, for example: Eu quero eliminar em casa o desperdício de plástico?

Yes, that word order is possible and still correct.
The most neutral order is:

Eu quero eliminar o desperdício de plástico em casa.
But:
Eu quero eliminar em casa o desperdício de plástico.
is also fine and might slightly emphasize em casa (at home) by bringing it earlier. It’s stylistic, not a grammar error.

What exactly does em casa mean? Is it the same as na casa?

em casa means at home / in the home in a general, non-specific sense. It doesn’t use an article.
na casa = em + a casa and means in the house (a specific house, or physically inside the building).
So:

em casa – at home (where you live, in general)
na casa – in the (particular) house
In your sentence, em casa is the natural choice.

How would I say in my house specifically?

You can say:

Eu quero eliminar o desperdício de plástico na minha casa. – in my house (physically, in my home).
You could also say:
Eu quero eliminar o desperdício de plástico cá em casa. – here at home (very natural in European Portuguese; cá adds the idea of “here”).
Plain em casa already usually implies “at my home” if you are talking about yourself.

How do you pronounce Eu quero eliminar o desperdício de plástico em casa in European Portuguese?

Approximate IPA (European Portuguese):

Eu – [ew]
quero – [ˈkɛɾu]
eliminar – [ɛlɨmĩˈnaɾ]
o – [u]
desperdício – [dɨʃpɨɾˈdisju]
de – [dɨ]
plástico – [ˈplaʃtikʊ]
em – [ẽ] or [ẽj̃] (nasal)
casa – [ˈkazɐ]

Spoken smoothly, you’ll hear a lot of linking and reduction, something like:
[ew ˈkɛɾu ɛlɨmĩˈnaɾ u dɨʃpɨɾˈdisju dɨ ˈplaʃtikʊ ẽ ˈkazɐ]

How would I say this in the past or future?

Using the same structure, you just change quero:

Past (imperfect, I wanted / I used to want):
Eu queria eliminar o desperdício de plástico em casa.
Simple past (perfect, I wanted at a specific time – more formal/literary):
Eu quis eliminar o desperdício de plástico em casa.
Future:
Eu vou eliminar o desperdício de plástico em casa. – I’m going to eliminate…
or the synthetic future (more formal/written):
Eliminarei o desperdício de plástico em casa. (much less common in speech).