How do I get around Istanbul-Atatürk airport when flying to Izmir?

Don't panic! Istanbul airport is generally just fine (much nicer than most American airports, in fact) and in my experience quite friendly.

1.5 hours should be OK as long as your incoming flight is not delayed. If you do miss the connection, and you have a single ticket for the whole trip, you'll be put on the next flight to Izmir and there are plenty (hourly or so on Turkish).

I'm from India, but have flown often through Istanbul. Istanbul has 2 airports, check which one you are landing in and if your connection is from the same airport. 1.5 hours is fine, even in the bigger airport (Ataturk) and there are plenty of flights to Izmir if your incoming flight is delayed. I found the airport staff and Turkish people in general to be polite and friendly to foreign visitors of all nationalities.

If you look at Istanbul Atatürk airport on Google Maps, you'll see that it only has a single terminal measuring about 1.6km between the most distant gates - 1,5 hours layover is plenty of time.

So pretty much the only thing that can make you miss the connection is if your first flight is delayed. In that case, since you booked the flight as one, the airline has to get you to your destination, typically by putting you on a later flight, free of additional charges.

I've heard how terrible this airport is, how rude people are to the Americans, and seriously delayed flights.

If you look up reviews for any airport in the world on the Internet, you'll always find some horror stories by some wronged passengers. It's unfortunate, but it happens all across the world, and cannot be otherwise considering how many millions of passengers each major airport serves every year. Think of it this way: if an airline flies to Atatürk, they deem it reasonably safe and convenient for their passengers. And it just happens that over 60 airlines from all over the world fly to your destination.

What do I need to know to fly international as an American? I have my passport already. Its brand spanking new so that's good. But I just feel so under prepared for this trip.

You need not know anything in particular, but learning the basics of the language and the social etiquette will help, because not everyone will speak perfect (or any) English. This is not a requirement in any way, but it will help you mentally prepare for the environment you'll be in and not feel too alienated.

Lastly, try to focus on the good things in your travel and avoid applying negative stereotypes. Visiting a country with prejudice is a recipe for an unfulfilling trip and more problems along the way.