Quest with a Friend
Although The Elder Scrolls Online can be played solo most of the time, if you are looking to speed up your leveling, bring a friend along. This has two major benefits, and will generally improve your experience since you have someone to beat things up with!
Questing with a friend, or even a few people will greatly speed up how fast you finish quests, esspecially ones that involve killing multiple enemies. The faster you complete quests, the more you can do an hour to increase your experience points. Being in a group will also allow you to tackle harder quests which reward more experience. When soloing, you typically can't take on monsters that are more than a level or two above you, but in a group, you could be level 20, tackling level 25 quests for much bigger experience point rewards.
Getting married in ESO is an easy process, and the reward can is the Ring of Mara, an item that boosts all experience points you receive by 10%. Although it may seem like a small amount, it adds up very quickly. This bonus is applied to everything you do as well, meaning it will increase your per-kill experience and your quest reward experience. The latter is the most important, as quests reward huge chunks of experience at a time. Now you are getting a huge chunk plus another 10% and if you couple that with the strategy suggested above, you will be blasting through levels.
World Bosses, Dark Anchors, Dungeons
The world is filled with enemies to kill, all of which will reward you experience points. World Bosses that spawn at random times, and Dark Anchors that respawn rather quickly are great for making some easy experience while also obtaining loot. Questing will always be the most efficient way to get levels, but these offer a good alternative, and will reward you with plenty of items in the process.
Dungeons take this a step further. If you can get a group of four people together, you can run dungeons endlessly. If you go in with the Ring of Mara, you will receive a nice boost through the entire dungeon, and since dungeons tend to have multiple bosses, that experience point boost will add up nicely by the end.
Read All the Books!
Bookshelves are scattered throughout the land, and just like the previous Elder Scrolls games, certain books will reward you will skill-ups. This will not contribute to your overall character level, but it's an amazing way to level up individual skills, specially those that you don't use. It only takes a second to interact with a bookshelf, and you don't have to actually read it, you can simply interact then walk away.
Where this really shines is in the crafting department. Leveling up crafting skills takes time and money. Reading books however, is quite free. There is no way to predict what books you might find, or how often you will find skill books, that is why it is important to check every bookshelf. If you're lucky though, you will get some free skill ups in the crafting field you wanted to pursue, saving you time and more importantly, money!
Low Level Skills vs High Level Enemies
The amount of experience it takes to level a skill doesn't change based on your level or any other factor. So say you want to level your bow up and it is level 1, but your character is level 20. You can use a level 20 bow to fight level 20 monsters and gain experience a normal level 20 should. Since your skill is so low though, it will blast through the levels due to the high amount of experience that is being rewarded. Of course, the higher level enemies you fight the greater the effect will be, so a max level character can fight level 50 enemies, and level up a brand new skill very quickly.
While exploring, you may notice that you gain experience. This occurs every time you discover a new area on the map. The rewards may seem small, but there is armor in the game that can boost up the amount you receive each time you find a new point. Wearing a complete set of exploration armor can boost the amount by over 50%, which is no small joke. You won't gain dozens of levels through this method, but it is a great way to boost your experience as you roam around inside a new zone.
What About Crafting Skill Ups?
Well sadly, there is no sure fire method to level up quickly, however there are some items that can help speed your skills along, if only by a bit.
Items with the description of "Increases inspiration by extraction x%" will give you more experience when you break them down. They tend to be rare, but if you ask around, you can likely buy many unused pieces from other players. The boosts may be small, but just like the Ring of Mara, they add up over time.
edit: or... you could steal loads of stuff, dismantle it and level up that way, which is what i did. you dont even need to take the stolen goods to a fence... boom.