Do Muslim children take their father's last name?

Yes, usually. In Islam a child's lineage (nasab) belongs to their real father, and in the West that is normally shown by his surname. There's no required style of surname, and hyphenated names are fine. What Islam forbids is giving a child a false father, so stepchildren and adopted children keep their own father's name.

Lineage belongs to the father

When the Quran ended the old practice of treating adopted children as biological ones, it gave this instruction:

ٱدْعُوهُمْ لِـَٔابَآئِهِمْ هُوَ أَقْسَطُ عِندَ ٱللَّهِ "Let your adopted children keep their family names. That is more just in the sight of Allah." Quran 33:5

Before this verse, people called the Prophet's adopted son "Zayd ibn Muhammad". After it came down, they went back to calling him Zayd ibn Harithah (Sahih al-Bukhari 4782). The Prophet ﷺ also warned that whoever knowingly claims another man as his father, Paradise is forbidden for him (Sahih Muslim 63). The point is truth about who you come from, not any particular naming system.

How this looks in the West

In many Muslim cultures, people don't have a fixed family name. A child is known as "Yusuf ibn Ahmad" (son of Ahmad) or "Maryam bint Ali" (daughter of Ali). In Europe and the USA, the legal system uses a surname, and the father's surname fits the principle naturally.

Different parents' surnames: hyphenating or using both is fine. The child's connection to the father stays clear.
Mothers: in Islam a woman keeps her own family name and lineage after marriage. Scholars differ on whether using her husband's surname for convenience is merely discouraged or acceptable, as long as no false lineage is implied. If it matters to you, ask a local scholar.
Converts: becoming Muslim doesn't mean changing the family name. See Do I have to change my name after becoming Muslim?

Stepchildren

A stepfather isn't the child's father in Islam, and the child keeps their biological father's surname. That doesn't stop you from loving and raising them as your own. The Quran says Allah has not made adopted children into real children (Quran 33:4), but it never discourages kindness. At school or the doctor's, you can say "my stepson" and use the child's own name. See How should a Muslim treat stepchildren?

Adopted and fostered children

Islam encourages caring for children without parents, but it asks us to keep their family name and not erase their roots. In many Western countries, legal adoption gives the child the adoptive parents' surname. Where the law requires paperwork like that, speak to a trusted imam or scholar about how to handle it. A common approach is to keep the birth surname as part of the name, and above all to be honest with the child about their origin. See Can I adopt a child as a Muslim?

If the father isn't known

Quran 33:5 also says that if you don't know their fathers, they are simply your fellow believers and close associates. If you are in this situation, ask a scholar how to name the child, because the rules depend on the details.

Key points

A child's lineage belongs to their real father, usually shown by his surname.
Hyphenated and double-barrel surnames are fine.
Stepchildren and adopted children keep their own father's name.
Ask a scholar about unusual or legal situations.

Tip: If you're blending a family, write the children's full names, with their biological surnames, in a family document. It helps with school forms and family records later.