Anyone have any recommendations on best airlines out of Austin to Sapporo? Then off to Seoul and Beijing? Thanks!

A little more about your purpose/goals/duration of trip and price point might help.

I've flown JAL, ANA, Korean Air, they're all good.

You are aware that Austin totally sucks for traveling to pretty much anywhere? So, would you prefer to connect or depart from Houston or Dallas? There are several options to get you nonstop to Tokyo or Seoul.

The flightconnections.com site is always a winner for figuring out what airlines might get you where you want to go.

Years ago, I needed to go to Osaka and Tokyo and got a super cheap RT airfare between Orange County John Wayne airport and Tokyo Narita. I just took the train for the domestic travel around Japan. Check out the JR site for that.

China air is doable and can be pretty cheap. Food not the best. You know there is a 3 day visa free permit into China which if you are doing tourist works ok for Beijing.
Above is right for the JR(Japan Rail) pass. Well worth every penny. You actually can do Japan quite cheap.

It's been a few years but we flew up to Sapporo and then drove to Niseko to go skiing.

We flew into Tokyo / NRT. Our domestic flight was JAL, leaving from Haneda (also in Tokyo).

We flew business from IAH to PEK once which was "meh". No where close to Sing air

When heading East, it's Singapore or Cathay Pacific.

Skiing in Japan, exploring Seoul, then hiking the great wall. I am really wondering if it's worthwhile going business class? As the reviews haven't been great as far as I can tell.

Singapore Air is great to get to Tokyo, same with JAL. Stay away from United. If Seoul, fly Korean Air. They have a non-stop from Dallas and one from Houston...great flight! China Air is fine getting into and around China, from either Seoul or Tokyo. Chinese airports are a pain but certainly doable. We moved around in China with kids...and made it.
Haven't spent much time around Japan other than transfers and a couple of days here and there. Others seem to have a few good ideas on rail passes, etc.
Seoul is a BIG city. Frankly, not much to do that is different from any other big city. There is a ski resort a little south of the city. Other than that, the nightlife is pretty good and typical Asian. I would suggest going to Busan instead. Plenty of temples and sights to see outside the city and the city itself is fun - a lot of expat nightlife, a beach and plenty of good eats.
Bejing is awesome. I would spend most of my time there. So much to see and most is walking distance from Tiananmen Square. Markets are a good time, some of the best restaurants around, temples and museums, on and on. The Great Wall is unbelievable. Make sure and have your hotel set up a reputable tour, and try to get a private one. You will have most of the day at the GW. They will take you to a touristy-type place - either clothing or pottery or something like that, usually something a relative owns. No pressure to buy just part of the act and frankly, not too bad. Also, make sure and take the toboggan down from the wall after. I'll go back and try to find out but there is a preferred entrance (several to choose from). It is the one with the Subway shop there. I believe the only one with the toboggan run, too.
Also, from Bejing, you need to get to Xi'An and go see the Terracotta soldiers. That is a day well spent and a true wonder. The city of Xi'An is not too bad either, plenty of great restaurants though were happy with the hotel ones.

Flew Korean Air from Houston to Narita then into Inchon. It was military so of course they got me economy or whatever was cheap. Wasn't bad though. I'd fly it again

For coming to Seoul from Sapporo, look into Jin Air and Eastar Jet (Korean Airlines) and Peach Aviation (Japanese) if you want to go the budget rout. I'm heading to Sapporo next week from Seoul and managed to get some nice deals with Jin Air (granted it was around 4 months ago). Keep in mind that if you are coming towards the end of January, tickets might be a little bit more pricey due to Lunar New Years (celebrated in China and Korea).

A previous poster mentioned Busan. And while it is a great city to see in warm weather, the beaches aren't as nice in freezing temperatures. Plus, transportation in Busan isn't so great. It's kind of a long city, so it takes a while getting from one side to the other. Seoul's public transportation is AMAZING. There's tons of cultural things to do inside of Seoul and in surrounding areas. The food and the nightlife are amazing (and cheap!). If your time is limited, Seoul is definitely your best bet. I currently live in Seoul. If you want to know anything else about the city, let me know.