Is what I'm thinking doable ?
Yes this can be done & all you need is a wireless router that allows Guest Network
With that, your guests would be able to connect to your wireless router not your ISP box.
I am not sure if you really want to allow your guests to connect to ISP box & why but it doesn't seem like a good idea to me
If yes, do I need a specific router (I haven't bought it yet) ?
Most of the modern age routers have this feature be it Netgear, Dlink or Cisco Linksys etc.
If you don't want to spend money on a new router, depending on what kind of router you currently have, you can also try some firmwares that provide such functionality
And how do I need to configure my network so that the computers in the public network can't see those in the private one ?
What you are trying to do is called Wireless isolation & it shouldn't be more than few clicks to enable it.
Note : Above screenshots are taken from Netgear & may vary from one router manufacturer to other.