The space between atoms depends very much on the medium you are talking about. In solids the typical distance between atoms is about the same as the size of the atoms themselves. In everyday gases at room temperature and pressure the distance between molecules is many times their size, and in deep space you can get densities as low as one proton per cubic centimetre!
You can get a rough idea of the average separation $\ell$ between atoms by using
$$ \ell \approx \left(\frac{m}{\rho}\right)^{1/3} $$
where $m$ is the mass of an atom and $\rho$ is the (mass) density of the material. This can be compared to the size of an atom, which for all elements is about the same at $\approx 10^{-10} - 10^{-9}\ \mathrm{m}$.
Space is full of fields like the electric and magnetic fields. You can think of certain types of "vibrations" of these fields as virtual particles, but the common view of modern physics is that the field picture is more fundamental. There are fields for all of the elementary particles, and the fields are constantly fluctuating due to quantum mechanics. You can think of temporary ripples in the fields as virtual particles, which are responsible for transmitting disturbances through space. Real particles are quantised excitations (or vibrations) in a field which propagate long distances.
Matt Strassler has gone to great lengths to explain this point of view in his popular articles.
Frederic Brünner brings up an important point about virtual particles. Physicists use an approximation called perturbation theory to do most of their calculations (because the calculations are really hard to do without making approximations). Virtual particles are a convenient way to organise these calculations, but you should not think of them as physical objects like real particles. In a sense virtual particles are misleading. What they really represent are rapid fluctuations of the fields (what I called ripples before). At large distances these fluctuations don't matter except when they average out to a smooth classical field. For the interactions between atoms, and even most of the interactions between electrons and nuclei, the classical field is all you need.