When you take electrons off an atom, it becomes an ion because it has less electrons than protons - this is ionisation. (first) ionisation energy is simply the energy required to knock off one electron from an atom (though the definition is more specific with gaseous moles). You "knock off" electron/s from the furthest shell.
Each atom/ion contains a positive nucleus - where the protons are stored. Therefore, electrons surrounding the nucleus will have differing levels of electrostatic attraction to this positive nucleus - the furthest electrons have a weaker attraction than nearest electrons since the distance is greater.
As you go down a group, the shells (or energy levels) increase. This means the attraction between furthest electron and nucleus is weaker. It is therefore easier to remove because there is less of an attraction between the electron and nucleus - therefore ionisation energies decrease down a group - not increase as you have said.
As you go across a period, the amount of shells remains fixed. However, the number of protons in the nucleus is increasing. This enables the nucleus to "pull" the furthest shell nearer, increasing the attraction between furthest electrons and nucleus. It is therefore harder to remove electrons and so the (first) ionisation energy increases.
If you want, I can explain why there are some dips in trends too