I think Acceleration might be the easier one to answer so I'll start with that. The truth is both! Humans can experience acceleration (pilots can black out from "experiencing" 4+ Gs). If I have acceleration and so do you I may measure your acceleration to be relative to mine; you may experience 4G's but you may only "have" 2G's relative to me. Humans do not experience velocity; if you were in a box traveling at 100km/hr you wouldn’t actually know you were moving unless you were able to measure your velocity with respect to something else (visually). But then you would have to ask yourself,
“Am I moving or is it?”
As for objects I believe they fall under the same category as humans although a rock will not black out, but the individual atoms will experience acceleration.