How do the first ionization energies of main group elements vary across a period and down a group?

They increase across each period, decrease down a group. As you go across a period the number of protons and increases. The positive nucleus then has a stronger attractive force on the electrons so it takes a larger amount of energy to remove an electron. As you go down a group the atoms are larger so the attractive force is weaker and it takes less energy to remove an electron.