sorgen für
to take care of
To look after someone or something; to take responsibility for someone's well-being or care.
Die Eltern sorgen für ihre Kinder.
The parents take care of their children.

to take care / worry
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to take care of
To look after someone or something; to take responsibility for someone's well-being or care.
Die Eltern sorgen für ihre Kinder.
The parents take care of their children.

The verb 'sorgen' (to take care / worry) is the basis for the noun 'die Sorge' (worry/care). Depending on the context, 'die Sorge' can mean either 'Kümmern' (taking care, as in 'die Sorge für die Kinder' — care for the children) or 'Angst/Unruhe' (fear/unease, as in 'sich Sorgen machen' — to worry), just like the verb.
Seine größte Sorge ist, die Prüfung nicht zu bestehen.
His biggest worry is not passing the exam.
The meaning of 'sorgen' (to take care / worry) changes completely depending on the combination with prepositions and the reflexive pronoun. Always pay close attention to whether 'für' (for) or 'um' (about/for) is used and whether 'sich' (oneself) appears with it.
Er sorgt für seine kranke Mutter.
He takes care of his sick mother.
The construction 'sorgen für' + accusative means that you take care of someone or something and assume responsibility. It has a positive, active meaning.
Ein guter Gastgeber sorgt für das Wohl seiner Gäste.
A good host takes care of the well-being of his guests.
The reflexive construction 'sich sorgen um' + accusative always means that you are worried or concerned. It describes a feeling of unease or fear.
Die Großmutter sorgt sich um ihren Enkel, wenn er spät nach Hause kommt.
The grandmother worries about her grandson when he comes home late.
The phrase 'sorgen für' can also mean 'to cause'. You can tell whether it means taking care or causing something from the object. 'Für die Kinder sorgen' means taking care, while 'für Probleme sorgen' means causing problems.
Der laute Streit sorgte für Unruhe in der Nachbarschaft.
The loud argument caused unrest in the neighborhood.
to care / concern / languish
'sorgen' and 'kümmern' can both mean that you take responsibility for someone or something. 'Für jemanden sorgen' often sounds a bit more general: you make sure that the person is doing well. 'Sich um jemanden kümmern' puts more emphasis on actually doing something, for example helping, organizing, or providing care. The difference between 'sich sorgen um' and 'sich kümmern um' is also important: 'sich sorgen um' means that you feel fear or unease. 'sich kümmern um' means that you act actively. That is why learners can easily confuse the words.

Die Eltern sorgen für ihre Kinder, auch wenn sie viel arbeiten.
The parents take care of their children, even though they work a lot.
Vor dem Termin hat sich Mehmet um die Unterlagen gekümmert.
Before the appointment, Mehmet took care of the documents.