“I’m fine, thank you” can really mean two things in American English:
1. Your English is not great.
2. You are actually NOT fine; there is a problem. Strangely, people say “fine” if they are being unfriendly or are upset about something. 🤷🏻‍♀️

So greetings…. Here are your options:

“How are you?” 
You can definitely say that!

You can also say, in a less formal environment:
How are ya?
How’s it goin?
What’s up?

Appropriate answers: 
I’m doing great. Thanks for asking. You?
I’m doing well. How are you?
I’m alright. 
Good. Thanks! You?

We don’t go too deep with these, but at times it’s appropriate to add a bit more detail.

For example: 
I’m doing well, but I’m a little stressed/tired right now. 
I’m great. It’s a lovely week.
I’m alright. I’ve been sick this week, but I’m getting better. 

Now when you’re changing the way you say something, you need to PRACTICE to get comfortable. As a Turkish learner, I had to say “yaaa” 1 million times by myself before I felt comfortable saying it to someone. 🙂 

Use this shadowing exercise to repeat after me. 

Directions:
Repeat after the speaker. (There will be a pause after each statement from the speaker. When the speaker pauses, repeat what she said and try to sound as much like her possible.)

We are going to practice greetings by shadowing. Repeat after me, and try to say it exactly like I say it. 

Let’s start with “how are you?”

If you are the first person to greet:

How are you?

If you are responding: How are YOU?

How are ya?

How’s it goin’?

What’s up?

Responses:

I’m doing great, thanks for asking.  You?

I’m doing well, how are you?

I’m alright.

Good, thanks! You?