Lapis Lazuli is AFAIK mainly for decoration. Iron, gold, diamond is for crafting tools/armor and so on.
Lapis lazuli is for enchanting, which is a worthy game element to pursue, I'd recommend reading up on it. It does great stuff. Need a diamond pickaxe to be able to mine the obsidian to make the table, and need lots of sugarcane for paper and cow leather for books and bookshelves to buff the enchanting.
Don't use gold to make equipment. It's worse than iron, on purpose, because that's how gold is as equipment in real life. It's used in at least one tool, like the watch I think, and has other minor uses here and there.
OMG!! That's very little lucky. I haven't seen any of those materials at all. Do you think every map that is randomly generated has its fair share of rare resources?
You're not finding a lot of them because you're not low enough underground. Every map definitely has guaranteed amounts of all the resources, there is at least one diamond deposit every 16x16 chunk. With the maps being 3072x3072, that's 36,864 guaranteed diamond deposits. And everything else is more plentiful than that even.
The best thing to do is go to a Y depth of 11 or 12 (look on your map) and dig in straight lines of 3 or 4 height. Move 2 spaces over, dig another line, in order to maximize the number of blocks exposed and minimize digging.
You can only find diamonds at that depth. You will also find redstone, lapis lazuli, gold, coal, and iron, but iron is rarer that deep.
You will run into lava at that depth, watch out! Dig carefully and be prepared to back up fast. Make an iron bucket or two, fill it with water, and pour it on the lava. If you can get it to touch source blocks it will turn to obsidian. Otherwise it will just turn to cobblestone and the lava will still be flowing behind that. You'll figure it out after some experimentation.