I started a new world and found a cave with a lot of coal and iron and I want to know how to get gold and diamonds faster.
Go to a "mesa biome" or as such a clay desert with mountains is called again and look for a cave or better a gorge or something similar.
For diamonds, all you have to do is look at levels 5 through 12. Most diamonds are found at these heights.
There's usually always a lot of coal and iron.
Xray and automine are your friends. Simply let the PC with MC run overnight with activated "automine diamond_ore" and look forward to dozens of double chests brimming with slides in the morning.