Interesting question. And I like the "Bodyweight Conditioning" article posted by
@pet' above; there are some good points in it.
But specific to the "energy" levels (energy systems?) and breathing....
I think of endurance as a mix of energy systems. When I go for a very long bike ride, I'm using alactic power, glycolytic fueling -- but
mostly aerobic fueling. If I can ride for 1-6 hours (it's been a while since I did a "century" ride of 100 miles in 5-6 hours, but I have done them in recent years), then my body must be good at using
both slow-twitch and fast-twitch fiber to produce power, recovering and refueling between harder efforts, and having a tolerance for continuing physical activity for a very long time. I also must have a strong heart and lungs that can continue a big output for hours on end.
Conditioning I tend to think of efforts of 5-30 minutes of higher (i.e. closer to max) output, and mostly fast-twitch fiber. This might be a lactate threshold bike ride, a CF 'metcon', a kettlebell snatch test of 5 or 10 minutes, a kettlebell circuit, a HIIT circuit, a 5k running race.... etc. All three energy systems are being used in this too, but but it's heavily glycolytic. All fitness aspects can help with these, but a strong aerobic base will help
more with a running or cycling race, and strength will help
more with a metcon or circuit requiring power or strength.
As far as breathing, I would say that it's generally not a limiter on endurance - you breath a lot harder than at rest, but are not maxed out. It can be a limiter on conditioning, huffing and puffing and having to take breaks during the activity to "catch your breath."
I prefer to spend most training time on the long-term building blocks of conditioning like aerobic base and strength. I can build short-term glycolytic conditioning with a couple of weeks of high-intensity work when I need