Welcome to the world of the server owner - It takes a copy of BUKKIT and a huge load of effort and dedication to set up a public server that is open enough for people to want to play on, and secure enough to discourage griefers and cheaters. I've been running my own server for a couple of months and it has been a HUGE effort. Just when you think you have got it all set up ok, Mojang updates the clients and all of a sudden, no-one can play on it any more until you update it.
Unfortunately it takes a while for the bukkit developers to re-code the bukkit server and even longer for the plugin authors to update their code. Also, when the terrain generation algorithms change, this can result in subsequent chunks not matching up and you end up with huge clipped areas that look awful.
Also you will find your players having a lot of demands on you all the time. But it can be huge fun to see what other people build.
At the very least you need the bukkit server from here http://wiki.bukkit.org/Setting_up_a_server you can copy your current world into it providing you know the seed.
You can then start adding and configuring plugins - this is a simple case of downloading the jar files from bukkit dev and dropping them into the plugins folder. At the absolute least you need essentials and permissionsbukkit. This will give you plenty of control to start with. Then look into WorldEdit and Voxelsniper - two excellent plugins for building and terraforming. After that - you can add whatever you want ie iconomy, mcmmo/heroes, ranks/roles/jobs etc
The players that are digging directly down to diamond are using an x-ray mod or texture pack on their local client that makes dirt and stone transparent. There are some block obfuscator plugins that prevent this by changing the block data, but I have never trusted them - they always have comments to say they are buggy, or cause other issues. The best solution to this problem is to build a (essentails) jail for these players and throw them in it permanently, or ban them from your server. You can then use WorldEdit to regenerate the chunks they have mined with the /regen command.
Running your own server can be fun, empowering and very absorbing, but don't let server admin take over your enjoyment of the game - remember why you started playing minecraft in the first place!