An actual subscription to Spotify lets you play music offline, though online it's all through the app rather than web.
I have Sonos which I really like – they have a gadget for connecting up existing hifi systems. It's spendy, though probably not so much in typical hifi component terms (the Ikea bookshelf speakers are £99, if you want stereo you have to buy two, of course). You can stream to speakers independently or group them. You can either play via the Sonos app and the NAS, any music service (Spotify, Apple, Google, Amazon etc) or internet radio, or via Airplay. The speakers sound pretty good to me (I have a bunch of Sonos Ones and Play-1s, and my wife has one of the Ikea lamps). Bose and others do similar multi-room systems.
Chromecast would also work though I'm not sure how many devices have it natively, it's usually a dongle for existing devices then a phone or similar to 'cast from.
For a cheaper route, there's always Bluetooth – adapters for the line-in of any stereo system are dirt cheap (and of variable quality since a DAC chip is involved) and there's no shortage of Bluetooth speakers on the market for the kid's rooms.