What is the quickest/easiest way to obtain gunpowder?

Aside from using server commands, there are only four ways to get gunpowder:

• Creepers
• Ghasts
• Dungeon Chests
• Witches

Of those four, creepers would be the easiest way to reliably (and safely) get gunpowder. Your best bet would be to make a mob farm (also called mob traps or mob spawners). The general idea is to set up a dark room for mobs to spawn in, and have them funneled into an area where they are either easy to kill or die due to the environment or traps.

This question covers how to make mob traps, which I also linked above.

The easiest way is to build a mob spawner and just afk overnight. Otherwise yes, I wish you happy creeper hunting.

What I saw from a video is that the best way to get string/arrows/bones/gunpowder is to make a mob trap. To make a mob trap you need to have a sealed, very dark and very high up room with no torches and no doors and you also have to have water flowing into a hole in the middle, the hole has to be deep, at the bottom you have to have a bit built out that's got lava and signs to hold the lava, once you've done that then you build a bit out and put water down so that when a mob/mobs spawn they fall down the hole and the water pushes the mob/mobs towards the lava so the mob/mobs die and the item(s) get pushed towards a point ready for you to pick them up.

Assuming you don't want to build a mob farm (which is some type of a large area filled with dark bits where all monsters--including creepers--will naturally spawn and then be funneled to a central area and killed automatically) you need to find a way to hunt creepers easily.

My method: Dig a 4x4 pit that is 2 deep. Wait until night. Lure creepers until they fall into the hole. Carefully kill them with a sword from the edge of the hole.

This is dangerous, naturally, but I generally don't wear armor and don't go gunpowder-hunting with anything in my pockets that I can't afford to lose.

Mobs are a bit smarter now, and sometimes don't like to wander off of cliffs. To encourage them to fall, you can place signs with their backs against the dirt blocks on the inside edge of the pit. Creatures mistake the tops of signs for something they can walk on and fall when they attempt.

Many others go out hunting at night with torches, food, dogs, and the best weapons and armor they can get.

Many people I respect swear by using TNT to mine quickly. In my experience it's more fun, but not any faster or more efficient (especially after factoring in the time and materials spent in the hunt) unless you've got a mob farm and thus a nigh infinite supply of TNT. I use TNT only when I gather it incidentally, and then primarily for making large pits or flattening things.

I think the easiest way is to kill time until a creeper spawns during nighttime, lure it far away from your build if you have one, and kill it as quickly as possible. This is the most efficient, but also most risky way to do it. Another way is to build a high darkroom, a low floor, and a tunnel between them. That way, if any mob spawns, they have a chance of falling through the tunnel, dying, and giving you their drops. Plus, you get other mobs' drops as a bonus.