if you go into CMD and type
it will send an echo (a 'ping') to google and tell you all the routers the packets travel through to get there.
The private range addresses within a LAN are
10.x.y.z (Class A private range)
172.16.x.y up to 172.32.x.y (class B private)
192.168.x.y (Class C private).
Public IPs can be any combination of classful dotted decimal numbers, up to a maximum number per octet of 255.
Most home LANs use class C addressing since it has a greater number of subnets and fewer hosts per subnet, since home LANs dont usually need more than 1-5 hosts per subnet. Most medium companies use Class B addressing for an almost identical number of hosts to subnets (allowing around 16,000 hosts i think?). Class A has so few subnets (254 compared to 16m on class C) and so many hosts, it is rarely used.
The internet is a big mesh of routers connected together in what we would call a WAN. When you send packets to a URL, the packets 'hop' from one router to the next until it reaches the destination, or it's maximum number of allowed hops (TTL)
giving someone your private IP is useless, they'd need th public IP of the router. Anywaym IPv4 addresses ran out about a week ago. The transition to IPv6 should be well underway. Private will become Link Local (FE80) i believe.
as FlightSim said, it's made very difficult to get router and home IPs maliciously, since all sorts of firewall technology, masking and IPSec technology exists to get in the way.