The periodic trend for atomic radii is that they increase as we go down the column and decrease as we go across the row. For first ionization energy, the trend is to increase as we go across the row and decrease as we go down a column.
Both of these trends are related to the effective nuclear charge (Zeff). Effective nuclear charge is the actual charge felt by an electron in an atom. As the atoms get bigger, there are more electrons between the nucleus and the outermost electrons which results in the ionization energy decreasing. It's easier to remove the outer electrons because they are not feeling the full force of the protons because they are shielded by the inner electrons.
When we look at atomic radii, we see the same idea for going down a column. As there are more and more electrons and the electrons are being added to orbitals higher in energy, then the outermost electrons are not controlled as much by the nucleus so they spread out even more. The shielding which reduces the effective nuclear charge increase with bigger atoms.
For the radii decreasing as we go across the row, the effective nuclear charge isn't increasing significantly because there are only very small increases in shielding when we add electrons to energetically equivalent orbitals. As a result the increasing charge in the nucleus can pull those outer electrons closer. As we go across the row, we see increasing numbers of protons and electrons so we increase the force between them (Coulomb's law) but we don't increase the shielding.