Quest objectives in ESO are extremely varied so you can have a lot of fun and not get bored easily. Some quests will tell you to kill some NPCs, others will ask of you to find missing people or save them, disguise yourself to infiltrate undetected and gather information, acquire missing ingredients or items, talk to people and help them find their way and many other kinds of tasks.
Usually a quest has a chain of objectives and you'll have to complete all of them to receive your quest reward and unlock new quests. Quest objectives can also help with exploration as they often send you to new, undiscovered areas and guide you towards collectibles like skyshards and lore books.
Every character has 3 major quest lines: the main quest line, the Mages Guild quest line and the Fighters Guild quest line. The rest of the quest lines are local and limited to the current area you're in.
Sometimes quests will send you in dungeons where you'll encounter strong mobs and bosses. Completing a dungeon has its own benefits, such as decent XP gain from killing NPCs and an achievement.
Not all quests are PvE-based though. If you go in Cyrodiil, ESO's PvP area you will receive some quests there and when you complete those, you will obviously receive PvP rewards, besides gold and experience.
The main quest line will present a new quest every 5 levels, but be careful. Some of the objectives involve defeating powerful NPCs which can be extremely difficult even if your character is several levels above the quest level. Most of the times, the NPCs can be defeated by properly using interrupt, blocking and dodging mechanics as they have powerful abilities that can quickly take you down unless you do something about it.
In these situations it's OK to flee and return with better gear and after you've invested a few more skill points in your abilities to increase your winning odds.
Quests that are 2-3 levels above your own can also be very difficult to complete if they involve fighting NPCs. In ESO NPCs are powerful even when they match your level, and even more if they're above it. Sometimes two NPCs will work together to take you down, combining their abilities and making your job harder. Usually the caster needs to be killed first as he has the highest damage/healing abilities even though he dies much faster than the rest of the mobs.
Available quests don't show on the map (only quest objectives) but you'll see a mark above the NPC's head once you get close to it. Sometimes you'll have to go to them and other times the NPC will run to you asking for your help.