Wait, then similar to my starts.
I've come up with a few different methods since this post, but figured I'd elaborate.
What I tend to do, is first dig into the side of a hill like before, set up a couple chests, a crafting table, make a spot for a bed as usual.
But instead of digging a staircase down until I find a cave, I'll instead go out and look for someplace to find some coal, or if I absolutely must, I can also burn some wood to get charcoal. Whatever I have to do to get some torches.
Then it's to get a couple hundred sticks and make about 120 ladders and start digging a vertical shaft. Not by digging straight down, everybody knows that's suicide. No, I'll stand one block to the side, and dig 3 down, and then jump into the hole, and dig 2 blocks so I can step one block to the side, put a torch there, put the ladders on one wall and keep extending the shaft down.
If there's nearby sand I'll get some of that too in case I hit a huge void, I can just drop sand to make the shaft to attach the ladder to, but otherwise, it's straight down to -62-ish to make some diamond mine strips. Hopefully on the way down, find some caves to hunt for iron as I'll need at least one iron pickaxe (I'll only use it if I come across gold, diamonds, or redstone, otherwise use stone pickaxes on the rocks). It's slow at first but it usually doesn't take long to find the first 3 diamonds and from there, just keep making shafts 2 blocks apart from each other until I have full diamond equipment.
After that? uhhh...
That's usually where playthroughs eventually end because I can't really find much else to do. I mean, I've been to the Nether a few times, but.... never really saw the point of going there. Found one bastion and even found a nether fortress or two, but they mostly have the same crap you'll find on the surface. I know there's ancient debris there, but those are annoying to find and take absolutely forever.
I could maybe try to kill endermen to get eyes and get to the End, but eh. I've killed a bunch of them and only got like one or two eyes, so that seems like another boring grind.
Still come to the conclusion that the game is absolutely devoid of any real meaty content. It's basically a sandbox without toys. There's lots of sand and... very little else. Maybe one or two sand molds.
Compared to Terraria, which is also a sandbox, but also a whole box of toy cars, sand molds, figurines, and lots of other stuff to play with too, it just doesn't have much going for it other than the fact it's 3D. There is surely more they can do with it other than adding yet more passive critters that drop nothing, and don't really do much of anything special and/or adding more useless materials that only make 1-2 items you don't care about (Copper is so annoying...)