"My name is Kaleo."
Translation:ʻO Kaleo koʻu inoa.
That has actually nothing to do with the beginning of the sentence. 'O stands in front of names (places, people ...) and e stands in front of commands or in front of names when you address people. (E hele e Kaleo! = Go, Kaleo!/E Kaleo, e hele! = Kaleo, go!). So when these names or commands stand at the beginning of the sentence, the 'O or E stands in front of them.
There isn't a simple, general answer. I'm going to talk about "‘O" in sentences like this one.
Beginning Hawaiian is often taught with an emphasis on "sentence patterns". There is a sentence pattern in Hawaiian called the "‘O equational pattern". According to http://www.donch.com/lulhconstr.htm
"‘O Equational Pattern sentences balance a noun or pronoun with another noun or pronoun and, as in the He Pattern sentences, there are no verbs used."
"He Pattern" sentences start with "He"; "‘O Equational Pattern" sentences start with "‘O". Both "he" and "‘o" also show up in non-equational sentences, and can appear in other places in the sentence.
The equational sentences have no explicit verbs. (They implicitly have a form of "to be".) The problem is that Hawaiian has a lot of stative verbs that look like adjectives to English speakers, so there are sentences that look equational but aren't.