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.

Any mob trap within 16 blocks of the bottom of the map will catch slimes as well, but you have to be careful to not hurt the slimes too much to get them to split the most. Punching and drowning are the safest ways to ensure that they'll split into more slimes. Also, you want a large room so that bigger slimes will spawn which can then have more splits.

The most important thing besides the depth is that it's in a chunk suitable for slimes. Slimes can only spawn in 10% of the chunks, picked randomly depending on the seed. You can calculate which chunks are suitable with this tool by trunkz though: http://www.minecraftforum.net/topic/397835-find-slime-spawning-chunks/

They only spawn a tenth as often as other mobs, but they'll also spawn in any light level, so if you're digging for diamonds at bedrock level, you'll usually find quite a few naturally.