initially

"Initially" means "at first". It's a formal word, so you use it in essays, formal speeches, and so on. "Initially" is used in phrases like these:

They initially refused to talk to the police.

It was a lot harder than I initially thought.

We initially considered staying in Denver, but neither of us was able to find a job there.

You can see that "initially" is used in front of a verb, which means that "initially" is an adverb.

(someone is) hesitant to (do something)

Being "hesitant to" do something means that you wait because you're not sure whether you want to do it or not. It's similar to being nervous, but "hesitant" also includes the idea of waiting before you decide.

Here are a few other examples of being hesitant to do something:

I think it'll probably be OK. I'd be hesitant to say that for sure, though.

Parents are usually a little hesitant to leave their kids at home alone at that age.

(someone is) persistent

Being "persistent" means that you continue to try something again and again, even after you've failed a few times. In the example above, the speaker probably asked his future wife for her phone number several times before she gave it to him.

In English, "being persistent" has a positive image. People who are persistent are thought of as hard workers and are often successful in life.