What are the fastest massive (not massless) objects a regular person (i.e., someone not working at a particle accelerator) will encounter? Of course, massless objects -- photons, gravitons -- move at the speed of light (299792458 m/s).
Neutrinos -- recently proven to have mass -- stream through your body constantly. They travel very very close to the speed of light, though their precise actual speed has yet to be determined.
Dark matter also streams through your body. Unknown speed; some probably close to the speed of light.
The decay products of cosmic rays hitting the atmosphere also hit or pass through your body. I am unable to look up their speed, but many are close to the speed of light: the muons must be traveling at that speed or else they would have decayed before they got to you.
Obsolete technology, part 1: The electrons in a cathode ray tube (CRT) in a television or monitor travel about 90000000 m/s or 0.3c, impressively fast. CRTs are technically particle accelerators.
Next are a few items of thermal velocity. Velocity is proportional to (only) the square root of the temperature.
The air molecules (or ions) heated within lightning stroke are probably traveling pretty fast.
Does a small static electricity discharge cause high speed air?
Hydrogen molecules are light, so move faster at a given temperature compared to more massive molecules. An unburnt hydrogen molecule in the vicinity of a hydrogen flame might be traveling 5800 m/s or 0.00002c.
Obsolete technology, part 2: An argon atom inside an incandescent light, having just bounced off the 5800K hot filament might be traveling 1000 m/s.
Gas molecules at room temperature range from 300 - 2000 m/s.
Next, physical objects:
A typical bullet travels 350 m/s.
A cracked whip exceeds the speed of sound, 343 m/s.
A commercial jet plane cruising at 600 mph travels 268 m/s.
A car going 60 mph travels 26.8 m/s.