Consider going with a specialized managed hosting service for minecraft; they're fairly affordable and tend to scale in price based on amount of RAM the machine has. For reference, example of hosting companies are beastnode, creeperhost, and 'redstone host'.
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.)
It's fairly easy, feel free to email me if you have questions. Though I've noticed a modpack wich fancy things really adds to the experience (playing with my younger brother and sister) as in vanilla you're fairly limited when it comes to decoration and ways-to-go. Hosting a modpack trough technicpack [1] is basically loading the modpack and afterwards moving the folder to the server and running it over there.
