Learn Hardcore Swedish: Barnen vill helst ha pasta med sås istället för kött. - The children prefer to have pasta with sauce instead of meat.

Barn means child or children (it’s both singular and plural in the indefinite form, like “fish” in English).
To say the children, you use the definite plural:

ett barn – a child
barnet – the child
barn – children
barnen – the children

In this sentence we are talking about specific children (for example, the family’s kids), so Swedish uses the definite form barnen = the children.

Why do we say vill ha and not just vill?

In Swedish, vilja (to want) usually needs another verb in the infinitive, or ha when you want a thing.

vill ha = want to have / want (an object)
vill äta = want to eat (an action)

Here the children want a thing (pasta with sauce), so vill ha is natural:

Barnen vill ha pasta = The children want pasta.

If you only say barnen vill, you would normally expect something after it: barnen vill leka, barnen vill äta, etc.

What exactly does helst mean, and why is it after vill?

Helst means something like preferably, most of all, or rather (in the sense of preference):

Barnen vill ha pasta. – The children want pasta.
Barnen vill helst ha pasta. – The children would prefer to have pasta / would rather have pasta.

About the position: in main clauses, Swedish usually has the verb in second position (V2 word order). Adverbs like helst typically come after the conjugated verb:

Barnen vill helst ha pasta …
- Barnen (1st element)
- vill (finite verb in 2nd position)
- helst (adverb after the verb)
- ha pasta … (rest of the clause)

You could also move the adverb in some other positions for emphasis, but vill helst is the most neutral, natural order here.

Could I say Barnen vill helst äta pasta med sås instead of vill helst ha? Is there a difference?

Yes, you can say that, and it’s correct. The nuance is:

vill helst ha pasta – focuses on having/getting pasta as the preferred dish.
vill helst äta pasta – focuses a bit more on the activity of eating pasta.

In practice, both are very often used and usually understood the same way in this context. If you’re choosing what to serve, vill ha is slightly more common, because it’s about what they want to have for dinner.

Why is there no article with pasta and sås? Why not pastan or en sås?

Pasta and sås are used here like mass nouns, referring to pasta and sauce in general, not to one specific, known portion.

pasta – pasta (in general)
pastan – the pasta (a particular pasta everyone knows about)
sås – (some) sauce
såsen – the sauce (a specific sauce already known in the situation)
en sås – a sauce (one type/portion of sauce)

In pasta med sås we’re talking generically about the kind of food the children prefer, so Swedish uses the indefinite, article‑less forms.

Why is it med sås and not something like med en sås?

Med sås means with (some) sauce in a general way. When you’re talking about food combinations, Swedish often uses bare mass nouns:

ris med kyckling – rice with chicken
glass med choklad – ice cream with chocolate
pasta med sås – pasta with sauce

If you say med en sås, it sounds more like “with a sauce (one particular sauce)” and often suggests you’re describing or choosing between types of sauces, e.g.:

pasta med en krämig sås – pasta with a creamy sauce.

How does istället för work? Is it one word or two?

Istället för is a fixed expression meaning instead of.

i stället / istället – instead
för – for

Together: istället för X = instead of X.

In your sentence:

istället för kött – instead of meat

You can also move the phrase for emphasis:

Istället för kött vill barnen helst ha pasta med sås.
= Instead of meat, the children would rather have pasta with sauce.

Both i stället för and istället för are used in writing; istället as one word is very common in modern Swedish.

Why is it istället för kött and not something with än, like än kött?

Än is used mainly in comparisons (than):

Jag gillar pasta mer än kött. – I like pasta more than meat.

But instead of is not just comparing; it means in place of, as a replacement for. For that, Swedish uses istället för:

Barnen vill ha pasta istället för kött.
– The children want pasta instead of meat.

So:

mer än = more than
istället för = instead of, in place of

Is the word order fixed? Can I move helst or istället för kött around?

Swedish word order is fairly flexible for emphasis, but there are rules.

The basic neutral order here is:

Barnen vill helst ha pasta med sås istället för kött.

Other correct possibilities:

Barnen vill ha pasta med sås istället för kött, helst. (more colloquial, with a slight afterthought feel)
Istället för kött vill barnen helst ha pasta med sås. (emphasis on “instead of meat”)
Barnen vill istället ha pasta med sås, helst utan kött. (now istället directly modifies ha, and the meaning shifts slightly)

What you generally must keep:

The finite verb (vill) in second position in main clauses.
Adverbs like helst usually after the verb (vill helst is very standard).

So you can move some phrases for emphasis, but Barnen vill helst ha … istället för kött is the most straightforward version.

What form is kött here, and how would I say “the meat”?

Kött is a neuter noun:

(ett) kött – (some) meat
köttet – the meat

In this sentence we have istället för kött = instead of meat (in general), so we use the indefinite form kött without any article.

If you were talking about some specific, known meat, you would use köttet:

Barnen vill ha pasta istället för köttet.
– The children want pasta instead of the meat (e.g. the meat that is already on the plate/menu).