After flying Air Astana economy on two flights, I can say confidently that the airline knows how to treat passengers in economy. While I'm sure Business Class is great, I wasn't in it, so I won't speak to that. But it's clear that Air Astana wants to make a good impression on its economy passengers, which I cannot say for US carriers like United or American, in my experience.
At the end of the day, it comes down to the little things that were clearly thought out. The blankets provided by Air Astana, which were available on both flights that I took, were big, fluffy, and warm — not the thin, paper blankets of other airlines. The number of times a flight attendant came by to offer drinks, a snack, coffee, or tea made me feel like a valued customer on the flight, not just a number.
It helps a lot that Air Astana is a new airline. Nearly their entire fleet is planes that are around 10 years old or less. I was on a 767-300ER, but the majority of their fleet is Airbus A320 and A321. That makes a big difference when you are choosing your carrier. You have a much higher chance that you'll be on a fresh, well-kept plane with Air Astana, rather than one that's been through 20 or 30 years of flights.
The food was solid. It wasn't the best airline food I've ever had — that still goes to Japan Airlines — but the portions were generous and there were lots of snacks to go around. I particularly enjoyed the Kazakhstani sweets, which gave me a tiny window into the culture.
With all of that in mind, it should be obvious that I would fly Air Astana again. The only downside was transferring through the Almaty Airport, which was small, old, and had an excruciatingly long transfer process. But given how good the flight was, spending a couple hours in an airport that wouldn't be out of place in rural Montana isn't a huge price to pay. And if there are Air Astana routes that don't pass through Almaty, even better.
Give Air Astana a try if you have a chance. I don't think you'll be disappointed.