A notice on your first level says that health and damage increases with every level. By how much? Does it differ by class? And what else? Is it always one skill point per level? Do other stats change?
Running speed, reload speed and pretty much everything is constant, and does not grow on its own. It can be improved via Badass Ranks/Tokens by amount explicitly stated there.
Base health appears to follow the exponential function Health = floor(80 * 1.13^level) In other words, each level increases your base Health by 13% relative to the previous level. Equivalently, your base Health doubles every 5.67 levels. At level 50, your base Health is 36058.
Your damage does not increase itself, but there's a sliding scale that determines damage to targets of different levels. Thus, after you level up, you start to do more damage to a constant enemy of constant level. Here's the scale for BL1, I could not find one for BL2, maybe it's the same.
Also, "level up bonus is 100% more Damage for 60 seconds and full health."