The server is cheap enough, and if I wasn't so nuts it could handle a small number of players easily.
The client costs $26.95; you can buy it online with a credit card, or buy prepaid game license cards in some stores. You authenticate with Mojang's central server when you play the game. Actual game servers are usually configured to reject client connections if the user isn't authenticated. (You can turn that option off, but you get a big warning message in the server logs when you do.)
The same login for "vanilla" Minecraft also works with modded versions, such as those accessed through the Technic Launcher (http://www.technicpack.net).
While I myself can appreciate maintaining a linux VPS and installing things from scratch (erm, that's precisely how I do minecraft hosting), managed hosting 'just works' without a maintenance overhead, and there are easy to use web control panels for plugin installation and configuration.
Most servers that kids like to play on aren't standard - they are 3rd party modifications (ex: bukkit). Such 3rd party servers have plugins that make the gameplay more usable. Example of some useful plugins are essentials, worldedit, worldguard. The headaches you will typically get is upgrading the server (after a minecraft update) and installing plugins. The hosting services mentioned above make it incredibly easy.
My server is actually at my home, so my godsons and their mother use a VPN to log into it. It's easier to have it hosted on a cloud service like DO. (If you're just accessing it at home, you don't have to bother with the VPN part.)