OK, you purchase a skill.
Certain skills are passive, meaning you cannot activate them. Sedna's healing wind for example automatically grants health per second to every allied unit in range. Regulus' sniper scope ability increases his range, and there is nothing you need to do.
There are 8 "normal" demigods who have 4 activateable abilities. Using these abilities is relatively easy. Most require you to choose a target after clickng them (fireball, penitence, pounce) but some just go off like surge of faith or circle of flames or fire nova etc.
The other 2 have form switching, 3 abilities in mode 1, 3 more in mode 2