I do understand that open water and open ground cools by the means of convection — lower air takes the heat and goes up, where it cools.
But why does the Earth lose energy and where does it go? Does it emit heat radiation to space? Does the heat go underground? What does happen to it?
It emits infrared radiation to space.
This happens constantly, not only during the night, but during the day the net energy flux is positive because the amount energy coming from the Sun is much higher.
Things are complicated a little bit by the atmosphere, meaning that most of the radiation emitted from the ground does not reach space immediately, but instead gets first absorbed by the atmosphere and then re-emitted to space.
Also, some of the energy is first transported to the atmosphere by evaporation and condensation before being radiated away by it.
If you want to know more, this introductory explanation by NASA looks good.
In the picture (taken from the NASA website I linked above): satellite map showing the distribution of thermal infrared radiation emitted by Earth in September 2008).
The Earth's surface, and the atmosphere, is in thermal contact with outer space, with the heat transfer mechanism being radiation. The Earth-atmosphere system is constantly losing heat by this mechanism. Indeed, this is the predominant mechanism that sets an atmosphereless planet's surface temperature: a planet of radius $R_\oplus$ without atmosphere and lit by a star with light of intensity $I$ will reach thermal steady at temperature $T$ state when incoming and outgoing radiative heat fluxes are equal:
$$\pi\,R_\oplus^2\,I = 4\,\pi\,R_\oplus^2\,\sigma\,(T^4 -T_{CMBR}^4)\tag{1}$$
where the right hand side expresses radiative transfer described by the Stefan-Boltzmann law (neglecting the solid angle subtended by the Sun), $\sigma$ is the Stefan-Boltzmann constant the $T_{CMBR}\approx 2.7{\rm K}$ is the cosmic background microwave radiation temperature, the "temperature of outer space". On solving (1) for $I = 1200{\rm W\,m^{-2}}$ (the power whereby the Sun lights the Earth) we get $T = 270{\rm K}$. For Earth, a more careful calculation takes account of the Earth's albedo (i.e. the proportion of the incident radiation that is reflected straight off in to space and is never converted to heat - the temperature of a perfectly reflecting orb in space would be completely unaffected by incoming radiation and woulf neither absorb nor radiate, staying at whatever its initial temperature is). This lowers the result. Another source of discrepancy is the greenhouse effect, which more than cancels the effect of the albedo of an atmosphereless Earth. But you can see that the result is in the right neighborhood for Earth's temperatures.
During the night, when there is no sunlight warming the atmosphere and the ground, the Earth's surface and atmosphere tends towards the temperature of the outerspace $T_{CMBR}$. The thermal inertia of the air and Earth's surface means that of course the nighttime is not long enough for the temperature fall to anything like all the way to $T_{CMBR}$ but, as you note, it falls some of the way and our nights are thus cooler.
Though they aren't quite as high resolution as might be ideal, the meteoblue website appears to give reasonable meteograms for stations around the globe showing the energy components...
(Sample energy balance forecast, copyright meteoblue)
On the radiation section (the bottom of the many graphs), you can see:
So typically:
During the day we get sunlight bringing in energy, with latent heat (evaporation) and sensible heat (conduction/thermals) also ramping up, dispelling some of the rapid energy gain. Then through the night, we really only have longwave radiation as the factor... a large portion of which is returned to Earth.
In more arid areas, there's less latent heat during the day (so heats up quicker), less LW down at night (due to less water vapor) (so cools down quicker), and so larger fluctuations overall. The opposite is true of maritime regions.
On cloudy days, there's less solar radiation, but more LW down. Daytime warms less, but nighttime cools less. So small fluctuations overall.
Some energy does indeed get conducted (and emitted) downwards into the Earth. But it must be a long-term balanced process (a mix of primarily diurnal and annual oscillations within the first few meters)... otherwise if the underground kept gaining energy overall, it would cause it to continue to warm up. Likewise, a very small amount of energy is actually provided to the surface from conduction (and emission) from energy actually emanating within the Earth (due to radioactive decay). However it's a tiny amount (Wikipedia quotes it as 0.027% of the overall budget).
This graphic and this associated Wikipedia article give a larger depiction of all the avenues of energy exchange going on within Earth's atmosphere and their typical scales. To extend that fully through the surface, you'd just need to add the small components of radiation/conduction into and out of the underground.
