Mining

Branch mining is the most efficient. There's a mathematical analysis on it in the official forums (the link is to an archive of the post).

Here's a brief description of branch mining:

• You dig down to the bedrock, because that's where you're going to find the most different kinds of stuff.
• One or two layers above the bedrock, so you don't have it interfering with your work, you dig a long tunnel, 2 high and 1 wide, which will serve as the trunk.
• Then, every 4 squares on either side of the trunk, you dig perpendicular tunnels extending out 20 blocks (or as far as you want).
• You leave 3 undug blocks between because veins are generally 2x2 horizontally, so you will see ore in the vein even though there is technically one row of unexplored blocks.
• When you finish that layer, you can start one a layer above, offsetting the branches by one block so that you are more likely to find veins you could possibly have missed the first time.

So essentially, a long hallway (the trunk) with perpendicular hallways (the branches) coming off either side.

██▒▒▒▒▒▒  ██ blocks you should mine
██▒▒▒▒▒▒  ▒▒ blocks you can see
▒▒▒▒██▒▒  __ blocks you can't see if you don't mine ▒▒ blocks (there's none!)
▒▒▒▒██▒▒  Repeat this pattern as many times as needed

From a head on view

Travel

As with any mine, travel is most easily accomplished using minecarts.

You can place a booster to get your cart traveling at max speed in a short distance so that you can make the trip up quickly, and you can even attach chests to mine carts to carry all your findings from the mine up the track very quickly.

Short of minecarts, make sure you're using stairs and ladders to their fullest potential. You don't want to be jumping up one block at a time all the way from your mine to your base.

Further analysis of Diamond probabilities:

Facts we need:

1. Diamond occurs in 0.12% of rock (unverified).
2. Using an efficient strategy we can expose 16* blocks for every 4 we dig (the ascii art doesn't account for two blocks you expose in front of you, so its actually 16, not 14). This breaks down to 4 blocks per dig. We're using Strix's method because diamond ore can appear in veins of 1, so you do not want to skip over any blocks. (if someone wants to do the math for the other methods, feel free to).
3. Diamond pick lasts 1563 blocks.
4. We're assuming worst case scenario of all rocks and no caves, as caves improves diamond probability due to increased surface area with no digging required.

Simple math after this point:

1563*4*0.12% = 7.5

So you can expect to find 7.5 diamonds per diamond pick in the worst case scenario with average luck so you will increase your pick count.

You want the floor of your dig to be at level 11 to avoid dropping into lava, which makes the ceiling at level 16, which is still inside the diamond zone.

Even if the ratio is 0.09%, you should expect 5.6 diamonds per pick, you'll still average more than you spend. The break even point is ~0.048%