Welcome to the guide which will hopefully answer your question if you can change your class in ESO or not!
When you have created your character and you want to change anything about it, this is usually possible even ingame. Aside from your clothes and your hairstyle or tattoos/body paintings or skins, you can also change your entire Appearance. This is possible via the Crown Store. In the Crown Store you can buy upgrades to change your Alliance, Appearance, Name, Outfit and Race.
However, unfortunately you can't change your class in ESO. Don't be disappointed though right now, because it can very well be possible that there will be a future update to the game where you can change your class. It used to be that you couldn't change your Alliance, for example, and recently it was made available.
This isn't as bad as it sounds. You will level up several characaters while playing ESO and you can have up to 18 character slots. The first character is hard to level, but once you get used to it it won't take as much time for your second or third character.The good thing about ESO is, is that even when you have to start over with a new character, many of the things carry over!
Things that carry over are: