I feel your original question may have many other questions embedded in it. I have therefore provided some background as well as helping you understand how you might choose your private IPs.
Public IPs are very tightly controlled resources that are in very short supply. Private IPs are used on internal networks and can be whatever you want, providing they are separated by a NAT boundary.
Different IP networks are connected together by network routers. Network routers are simply devices that know how to route your network traffic to its destination and how to route the reply back.
So for example, if you are using a laptop on a private network with an IP of 10.1.1.12 and you wanted to visit a website on 126.96.36.199. Your laptop would send it's requests to the router hoping the router knows a valid route to the target IP.
Public and private IPv4 networks are often separated by a Network Address Translation (NAT) boundary. A NAT boundary is simply a network device (often the router) that effectively hides the source IP. So in the case above the IP address the server at 188.8.131.52 would see is a NAT'd IP, not the original 10.1.1.12 IP.
You asked what your private IP should be. In order to determine what your IP should be you first need to determine what network address you would like to use and how many nodes you are going to need on that private network. Normally people do not bother with this process and simply go for an easy option. I will lay out both an easy option and a slightly more involved opportunity to learn below.
Pick whatever network address you want, e.g. 192.168.0.0/24. This will allow you to use any private IP between 192.168.0.1 and 192.168.0.254. Give your router an IP of 192.168.0.1 and ensure everything on the network uses the subnet mask 255.255.255.0.
More difficult option:
Pick whatever address you want, but increase the /24 value to reduce the number of available IP addresses to match what you actually need. For example if you chose 192.168.0.0/25, this would allow you to use any IP from 192.168.0.1 to 192.168.0.126. Give your router the same IP, but this time use a subnet mask of 255.255.255.128.
The above example uses CIDR blocks instead of classes. The class of a network is not that relevant as such. What is more relevant is the address range and subnet mask.
/24, /25 or other netmasks simply determine how many bits of an IP address are used for the network address. Thus leaving the remaining bits for host addresses. To understand this you simply need to convert the IP to its binary form, examples below:
11111111.11111111.11111111.00000000 = 255.255.255.0. This is /24 because there are 24 ones in the binary address.
11111111.11111111.11111111.10000000 = 255.255.255.128. This is /25 because there are 25 ones in the binary address.
The more ones you have for the network portion, the less digits you have free for host addresses. Thus the higher the netmask value, the fewer IPs you can have on your network. The network mask also determines how many networks you can have.....but I suggest we ignore that bit for now :).