How do you know how fast or slow to go while hiking at elevation (say above 3000 meters) on an instant per instant basis? Elevation, weight carried, steepness, surface underfoot (e.g., trails, cross-country over granite, talus) all affect your energy levels, so it's not as simple as saying N feet and X miles in one hour. How do I pace myself? How do I know whether to push myself or to slow down or stop; I want to avoid "hitting the wall" or become so exhausted I can't go on.
Disclaimer: I usually climb up to 3000m or slightly beyond that.
I use a heart rate monitor. My pace is set by a heartbeat threshold. I try not to exceed roughly 150bpm. If I exceed my aerobic threshold, I know that I am working harder than normal. This can be sustained for awhile, but not for too long, especially where the air is thinner. You actually get more health benefits if you keep your heart rate relatively low for the long haul.
For estimating total time of ascent, I use 1 hour per 500 meters (if you are fit and used to altitude changes!) as a rule of thumb. Otherwise, you could adjust the h/m expectation to your fitness level/experience.
This is based on a mostly vertical climb. It of course depends on the angle and how much horizontal terrain you need to cover.
Where does 150bpm come from?
It is the highest recommended bpm for aerobic respiration (cardiovascular) activity before you go anaerobic (muscles produce more lactic acid) for a male of 30 years of age. Obviously you need to know your body, because 150bpm is a rough estimate based on averages that do not take into account the many factors of your body.
You can check the average for your age on various websites to get an idea of what your threshold might be. One such example that I did not necessarily use: http://www.aqua-calc.com/calculate/maximum-heart-rate
The OP asks:
How do you know how fast or slow to go while climbing at elevation (say above 3000 meters) on an instant per instant basis?
First, it sounds as though you do not have experience hiking at altitudes above 10,000 feet. If this is so, my advice is to take a few day hikes (carrying significant weight on your back) before embarking on a multi-day trip, especially if you have to meet distance goals on the trip, for example to get to good camping spots.
Your body will tell you if you are going too fast: listen to it. Your heart rate will increase and you will be out of breath to the point that you will be uncomfortable.
These are the same messages your body sends you at sea level when you are running or working hard, but they will come much sooner at altitude.
I recommend that you experiment with pace, with the goal not speed or distance but endurance; that is, how long you can go before your body insists that you stop. If you feel that it sure would be nice to stop, but that you don't have to stop, you've hit the right pace. At the end of the day, you should feel tired, but not wiped out.
I'm not going to discuss acclimatization or precautions against, or how to recognize, altitude sickness: that goes beyond the scope of this question.
Choose a steady pace that you can keep up non-stop for two hours. If you notice that you need to catch your breath earlier, you are walking too fast. After two hours it's time for a break.
The first time I went hiking with a group other than my family, we were fifteen Dutch youths (17–21). There was a steep trail uphill climbing roughly 800 metre, which should take around two hours at low elevation. When we started, almost everybody overtook me within minutes. Within 90 minutes, I had passed all the others one by one, all while they were standing still catching their breath. I was the joint first to arrive at the summit, together with a girl who decided not to speed past me at the start but stay just behind me, then thanked me for keeping a steady pace.
Listening to your body is good advise as others said: try to maintain relaxed, deep breathing, and when it becomes too much for you to do comfortable, you should slow down or stop to reset back to that. Relaxed in this case just means without tension or any struggle to breathe and get the air you need - climbing a mountain, breathing will be vigorous, but it can still be calm and under control. Someone else mentioned using a heart rate monitor - though I don't have experience with that, I imagine it can help like awareness of your own breathing.
Differing from other answers thus far, I would recommend keeping distance goals based on the overall checkpoints planned on that day. Practicing with simple day hikes and easier backpacking trips certainly makes sense, and even in those cases you will have checkpoints you plan on, whether it's a half way point, a peak, your car, a camp site. In more elaborate trips you will have an understanding of the terrain and natural landmarks to guide you along the trail - practice good orienteering by checking your progress based on terrain, landmark, and trail based checkpoints; consider if your goals are in harmony with your progress or if you need to adjust.
As you practice you'll get a feel for how to combine the above two paragraphs for you and even for your group so that you can make the best steady progress. You'll have some times you just stop to cool off and reset, but importantly you will begin to combine progress goals with your body's resting needs.
For example, I'll make steady progress up to a point I've pre-determined on the map since the last time I stopped to check the map (and maybe even planned before I embarked on the trip). I'll get to the point I expected to arrive at, usually a mile marker or some natural landmark, and I'll check my time and how I'm progressing through my day's planned checkpoints. All the while, I'm returning to a very comfortable breathing rate, I'm making sure I'm hydrated and ready to go to the next checkpoint.
In some cases, taking breaks as orienteering checkpoints also helps me recognize when I'm not on track, either because I'm not comfortable estimating my location to a spot on the map, or because I'm going too slow (or too fast) and should adjust my goals accordingly. Better to calmly realize and note these course adjustments needs ahead of time in calm conditions than later on with more pressure and potentially worsened conditions.