Couple things I wanted to add.. 'street' mining is best done with only 1x1 side-shafts. It's most efficient - you're revealing 5 blocks per 1 block mined vs 10 per 3 blocks mined.
When a friend and I do this we have one person mine the 2x1 'street' and the one behind covering the 5-deep 1x1 side shafts. Do this at level 12. When a cave or lava pool/sea is revealed, explore it briefly and continue (again just maximum exposure of blocks that can be diamond).
Also, I'd really recommend spending your first few diamonds on an enchanting table. Make a sapphire pickaxe (more economical than diamond as sapphire has fewer crafting uses). The sapphire pick has less durability, but you'll be enchanting it. Make a ruby sword when available, and some bookcases, and try to get a level 30 enchant. If you can manage to get fortune in any capacity on your pickaxe, it will greatly enhance your diamond gathering. Even if you don't get fortune, you'll still have a great (fast/durable/both) pickaxe. If you manage to get silk touch, that's great too as you can pulverize diamond ore for twice as much diamond.