I just got my enchanting table, and now I want to know how to get XP quick. My usual strategy is to go out and night and kill every mob I find, however I believe that there are faster ways that I don't know of. What is the fastest way, or just any way much faster than my way?
Depends on where you are in your progression.
One way is to the nether and just mine a lot of quartz, until I get up to level 30*. Once I have a silk touch pickaxe I then use that to get the ore, which I can then mine as needed.
Another method is once you have killed the Ender Dragon, it drops a lot of XP itself, but also lets you build a very efficient and fast enderman XP farm.
The other way I like to get XP is from cured villagers. They offer cheap trades and you get a 2 in 1, as you get XP from trading, and other useful items such as Mending books. Once you have enough mending books you can disenchant these at a grindstone for more XP. This does take some time to set up, however, and you will need a villager you can trade with to get emeralds, such as one buying paper.
Personally, I don't find mob grinders, e.g. dungeon ones, to be very efficient/fast, and I find it more interesting to explore the nether mining quartz as a go, especially on the latest snapshots.
* It is worth noting that once you go above level 30 you are essentially 'wasting' XP, as it takes more XP to go from level 30 to level 33 than it does to go from 27 to 30, so it is more efficient to do a level 3 enchant at level 30 and then get back up to 30 and do another than to get to level 33 and do 2 in a row.
You can create a mob grinder. There are several tutorials on the internet for easier or harder ones. It's should help you to get more XP but may require you to AFK if you want heaps.
The good thing about this way is you can go back to it when you want and if you have a drop collector you'll be well off. I'd reccomend making a skeleton grinder as bones can be used for white concrete and arrows a-plenty.