Increasing power to a pool boosts its regeneration rate by 50%, and reduces the regeneration rate of the other two pools by 25%.
Increasing weapon power boosts damage by 10%, while increasing shield or engine power reduces weapon damage by 5%
Increasing shield power boosts max shield capacity by 20%, while increasing weapon or engine power reduces shield capacity by 10%
Increasing engine power boosts max speed by 20%, while increasing weapon or shield power reduces max speed by 10%
All of the power pools have a "normal" regeneration rate and a slower "recently consumed" regeneration rate. Anything that consumes energy (taking a shield hit, using your engine boosters, getting hit by an ion railgun, etc) will cause the affected pool to switch to its "recently consumed" rate for several seconds (determined by the "delay" stat for each pool).
The default numbers for each are:
Weapons: Pool capacity 100, regen 10/s, recently consumed 4/s, delay 1.5s
Engines: Pool capacity 100, regen 5/s, recently consumed 2/s, delay 3s
Shields: Pool capacity varies, regen 5%/s, recently consumed 0/s, delay 6s
I haven't seen any bugs cross my plate about energy behaving strangely near the bottom of the pool, but if you have steps to reliably reproduce the glitch let us know.
The ideal thing to do at the bottom of the pool should be to stop firing, wait at least 1.5 seconds for the regeneration rate to kick into high gear, then open fire when you think you have enough energy built up to finish your opponent (although it may make sense to fire a bit early if you get a shot lined up and need to prevent the shield from regenerating).
Primary weapon accuracy and damage are linearly interpolated between the short, medium, and long distances (so it's smooth, not abrupt). Everything closer than short distance uses the short values.