Why do first ionisation energies decrease down a group but increase across a period?

The nucleus contains positively charged protons which attract the negative electrons, the first ionisation energy is the energy required to overcome this attraction and remove 1 electron As you go down a group the number of shells containing electrons increases. This means that as you go down the group the outermost electron is further and further away from the positive nucleus, it also has more electrons between it and the nucleus so it is therefore shielded from the positive charge of the nucleus. This leads to less strong electrostatic interaction with the nucleus so less energy is needed to remove the electron.
As you go across a period the number of protons in the nucleus (known as the effective nuclear charge) increases. However the number of shells with electrons in doesn't change. Therefore, as you go across a period and the nuclear charge increases the electrons will be more tightly held by the nucleus (greater electrostatic interaction) and so it will be harder to remove one - it will require more energy giving a higher ionisation energy.