You can change your local IP any time you want, just in your router settings.
Changing your public IP - well, since you're usually connected to the internet through a modem, that would involve changing the IP of your modem's counterpart at your ISP. Usually, they won't let you do that.
Changing the WAN IP in your router is easy to do (again, see the router settings), but all that would accomplish is that you won't be able to connect to the internet any more (see above - it wouldn't mean anything to your modem's counterpart at the ISP, which will listen to the world on its IP address while sending out requests from you on that address that you set in your router. Similar to you sending out letters with someone else as the return address. The letters will reach their destination all right, but the responss will arrive at whatever you wrote there as return address).