As we mainly only listen to Internet Radio we have a Bose Home Speaker 500 in the sitting room, paired to an additional Bose speaker in the kitchen, so only one playing device required.
Although in saying that, for playing CD's we have a Panasonic DAB radio/CD player in the conservatory, paired to various other speakers throughout the house. The Bose is quite fussy about what it pairs to (Bose only I believe), but the Panasonic is more than happy to pair with additional speakers (Sonos etc.) dotted about around the house. Each speaker needs to be within reach of your home wifi.
So, in short, you could buy a system and just add speakers for the other rooms. They can all be controlled by Alexa (the spy in the house
) if you so wish, or your smart phone, or in the case of the Panasonic, by using the remote. You could buy a Amazon Alexa Echo or Dot/Google Home and just add additional speakers (that's how we started).
If it's just for Internet Radio it's very easy to set up multiple speakers in different rooms from one device but if you want the CD function you will need a Radio/CD player with Wi-fi/Bluetooth compatibility.