Tempting as it is to measure velocities as fractions of the speed of light – and indeed to regard velocities as dimensionless as a result – or to identify velocities with hyperbolic rotations, I'll stick to the metric system.
The tectonic plates that make up Earth's crust move at speeds of order mm/year; one of which is nearly 32 pm/s. The average rate at which ocean-levels rose during the 20th century was around 50 pm/s (1.7 mm/year), and since the 1990s that speed has increased to around 95 pm/s (3 mm/year).
Human fingernails grow at about 0.1 mm/day, i.e. about 1 nm/s. Toenails grow at about a third of this speed. Around the age of fifteen, I suppose I grew taller by a respectable fraction of a foot in the space of a year or two; that's a speed of something like five nm/s. The ground at Yellowstone park moves up and down, as it floats on a lake of magma, at a speed of around 2.5 cm per year, or roughly 0.8 nm/s.
68 Martian sols into its stay on Mars, the Sprit rover was 300m from its landing pad; allowing for some days paused, that's an average of about 5 metres / day or 60 µm/s.
Pine Island Glacier, part of the West Antarctic Ice Sheet, is currently moving at about 3.5 km/year, which is 0.11 mm/s; this is considered fast. The ISS loses altitude, due to drag from Earth's atmosphere, at up to several hundred metres per day, i.e. several mm/s.
One metre per second is a fairly leisurely walking pace for a fit human adult. The furthest I've walked in an hour is 10 km, making for an average speed of two and seven ninths m/s; that was in a race whose winner took only 40 minutes, for an average speed of four and a sixth m/s. The famously difficult feat of running a mile in under four minutes involves an average speed of just over 6.7 m/s; while the small number of people who have ever run 100m in less than ten seconds have exceeded 10 m/s. An object dropped not much above The Earth's surface accelerates at just under 10 m/s/s (and just over the speed of light per year), assuming it's compact enough that air resistance is negligible; after it's travelled just 51 mm its speed is one m/s; 0.92 seconds later it's fallen 5.1 metres and is travelling at 10 m/s; by the time it's fallen 5.6 km, air resistance (or the ground) is sure to have intervened, since (by then, 33.8 seconds into its fall) it'd be moving at the speed of sound.
A Boeing 777 flies at about 0.27 km/s (600 miles / hour); parts of its engine spin at roughly twice this speed. The speed of sound in air, a.k.a. Mach 1, is 0.33146 km/s. When the Moon eclipses the Sun, as seen from Earth, the Moon's shadow moves across the Earth's surface at nearly 2000 km/hr, i.e. at roughly half a km/s. When an earthquake strikes, seismic waves radiate from the focus at about 4 to 6 kilometres per second.
The surface of The Earth, at its equator, moves at 0.465 km/s relative to (an inertial frame of reference moving with) the centre of The Earth; which, in turn, moves at 29.8 km/s relative to (an inertial frame of reference moving with the centre of) The Sun. Meteors strike Earth's atmosphere at about 60 km/s. The Milky Way and Andromeda are moving towards one another at a speed of 120 km/s (at their present separation, if they were in mutual circular orbit their speeds would be around 40 km/s each); their side-ways relative motion is, as yet, unquantified. The Milky Way's (flattened) inner stellar halo orbits at around 20 km/s, while its (spherical) outer halo orbits at around 70 km/s in the opposite sense.
The Sun, in its turn, moves at about 0.22 Mm/s relative to The Milky Way; and the local group moves at at about 0.6 Mm/s, roughly c/500, relative to the cosmic background of microwave radiation. The long-period variable star Mira moves through the interstellar medium at 0.13 Mm/s. The two (likely) black holes making up 3C 75 move through their local enbironment at a relative speed of about 1.2 Mm/s.
The speed of light is 299792458 m/s (i.e. just under 0.3 Gm/s); that's an exact integer now, thanks to the most recent redefinition of the metre. No physical thing, nor even information, goes faster than that: but the point where a laser pointer hits a distant enough surface, or a shadow falls upon some surface, may move faster, if the surface is far enough away and the direction from light source to object casting shadow, or the laser-pointer, turns fast enough. Similarly, when a distant star produces a burst of light, if it is surrounded by a disk of light-scattering matter at a suitable angle to our line of sight to the star, the light's echo off the disk may appear – to us – to spread through the disk arbitrarily fast.