As a long-time mountaineer, there are two specific things you can do, and one general thing.
First the general thing. Never walk so fast that you get out of breath and can't carry on a normal conversation without panting or gasping. You might think, if I just amble along, I'm never going to get anywhere. Well, if you rush it, you're not going to get anywhere either, and you'll get further if you take it easy. Taking it easy might mean that on a steep uphill at 10,000', you might take one step and rest for 5-10 seconds before you take the next step; whatever it takes. It's very difficult to recover once you get out of breath at altitude, and you waste a lot of energy.
The first specific thing is the mountaineers rest step. I said above to amble, but there's a specific way you amble. Your don't want to be like a slow motion film that keeps moving all the time. If you're on that steep slope at 10,000', you take a step forward, uphill, and then you pause when your foot hits the ground and relax the muscles in your front leg, while your weight is still on your back foot. This allows the muscles in the forward leg to rest for a moment, before you put weight back on them. You take a little rest with every step you take, and when you take a longer rest, you shift your weight from foot to foot occasionally so each leg gets rested. It makes a big difference in the course of a day.
The second thing is power breathing. Whenever you feel the effects of altitude, you don't just exhale. You blow the air out, with a puff like you were blowing out a birthday candle, which creates a backpressure in your lungs. You do this with each breath, and you do it all day, if necessary, so long as you feel any effects of altitude. The back pressure will enable your lungs to be able to take in as much oxygen as if you were 1,000' or so lower.
Oh, and it may be some small consolation that altitude sickness is one of the few things that seem to be less of a problem as we age. I don't know why; maybe we just don't make as many rapid unnecessary movements as we get older.