Hi Max, some basics about Sonos to start you off, some others on here are better experienced about Squeezebox.
Sonos can be wireless or wired or a mix of the two. For each room you want music in you need a zoneplayer, we have a quick product guide at
Sonos Product Guide / Multi Room Systems - Sonos Authorised Reseller. Sonos Authorised Installer to tell you about each zoneplayer type. At least one zoneplayer or zonebridge needs to be plugged into your home network/router (by ethernet cable).
Sounds like your router is in the kids playroom so in here you would have a Sonos zonebridge connected to the router (assuming you want to carry on in this room playing music via the laptop and Xbox rather than Sonos - dont waste a zoneplayer in a room where the router is but you dont want Sonos music). For the sitting room you would probably have a Sonos ZP120 zoneplayer (which has its own amp) into which you would connect speakers of your choice into the zoneplayers standard speaker connections. You could also consider the Sonos S5 zoneplayer which has in-built speakers to start with and is versatile enough to be moved from room to room. Either way the sitting room zoneplayer would connect to the zonebridge wirelessly or wired (ethernet), its your choice.
If you want to add more rooms of music in the future just add more zoneplayers.
Sonos is compatible with many internet music services as well as any download service that provides DRM-free tracks, iTunes as an example. It liberates the music that's stored on your computer (laptop in your case in the kids playroom) and makes it available to play in each room where you have Sonos. Plus, there are analogue line-in sockets on the back of every zonePlayer so you connect any external audio source with a line-out socket such as an MP3 player to any zonePlayer and you can also play that music in every room.
In terms of controllers, the Sonos CR200 is handheld, wireless and has a colour touchscreen. You can control any of your Sonos zoneplayers from any room with one or more controllers. Choose the music, see album artwork, track and artist information, build playlists etc. Every zoneplayer also ships with free desktop controller software (Mac or PC versions) and if you own an iPhone or iPod Touch there is a free app you can download from the Apple Store to turn it into a Sonos controller. You can mix and match controller types, just have one, or have multiple controllers (one per floor is a good start point).
Hope this helps as a starter.