You're confusing several different phrasal verbs here, Zoe. 'Look after' and 'Take care of' have a similar meaning. He looks after animals. He takes care of animals. Don't confuse these expressions with 'look for' (in your title) which means 'try to find' 'I'm looking for my keys. Have you seen them anywhere?' or 'Take care' ( without the 'of') which means be careful, 'Take care when you're driving this morning. The roads are icy.' Remember that you have to get phrasal verbs exactly right, otherwise they can have a different meaning.

look after animals (or children, for that matter!) - take care OF animals; care for animals - all of these are fine and mean the same thing. Take care WITH the animals has a slightly different meaning - it is a warning to be careful around the animals - as in take care with them, as they may bite.