Minecraft wiki entry for Mining, associated text: Requires a bucket (3 iron ingots), which you have to get by exploring some surface caves.
If you really can't find iron, you can alternatively utilize a natural water source, like lakes, rivers or oceans.
You need to be good with water (but come on, practice makes perfect!)
You need to dig in a specific way while going downwards, otherwise the water can get messy.
Alternatively you can dig the hole the old-fashioned way, and climb up using the water bucket.
Sprint-swimming requires a lot of food. (But food isn't that hard to get after all)
Depth Strider doesn't make it faster — it only speeds up horizontal movement in water, not vertical.
Therefore, if you are wearing Depth Strider boots, it's hard to maintain position in the water column, as a little bit of tilt can lead you off track substantially.
But this doesn't really matter, as you will likely be using a soul sand bubble column instead of this vetical shaft, by the time you have Depth Strider.