When they were kids I cared for them both equally.
When they were kids, I equally cared for them both.
is the most economical way to say what you mean, the first sentences will usually be understood to mean you had equal affection for them and treated them in the same way, the second sentence may be understood to mean you had equal affection for them.
When they were kids I cared for them both alike.
sounds slightly awkward to my ear, better might be
When they were kids I cared for them both in the same way.
Your final sentence
When they were kids I cared for each of them in an equal way.
sounds awkward using way since your probably mean the caring (treatment) was the same.