To get new skills in ESO, you have to kill monsters to earn spirit.
For each kill you will receive a certain amount of spirit. Once you have enough spirit you can spend it on new skills. Each time you gain a couple levels, new skills will be available for you to buy with spirit.
To access your skill tab press (K)
Ethyrs are special skills that can be equipped like gear. Once equipped, they provide the player with the skill. Up to four different Ethyrs can be equipped at a time. In addition to regular Ethyrs, there are also class-specific Ethyrs. Ethyrs offer players some very interesting abilities.
Class-specific ethyrs are generally given as quest rewards while regular ethyrs are monster drops. Strong monsters and difficult quests provide the most powerful Ethyrs as rewards.
After equipping an Ethyr you can open the Skill Menu (Hotkey "K") and find the skill in the Ethyr Skills section. Ethyrs can be leveled up by consuming Spirit to become more powerful. You can also click the Ethyr button to see all Ethyrs.
In the Ether Saga world, Nascent Skills provide a whole new level of customization. Once your birthday is selected, these skills become available as your character levels up. Keep in mind that your birthday can be changed by using certain items. Every race/birthday combination results in a different set of Nascent Skills. You get a new Nascent Skill at level 10, 30, 50, and 80.
For more information regarding Nascent Skills, as well as lists for race/birthdays, then check out the wiki page here