What is a WAN Address? How can I distinguish between a WAN and LAN address?
So in a IPv4 world a WAN address should be an address not in RFC1918 by the IETF. That means roughly speaking any address not in the range 10.0.0.0/8, 172.16.0.0/12 and 192.168.0.0/16 and of course some others, like the loopback subnet or the unassigned/ reserved space above 240.0.0.0/4. @Ron-Maupin has presented some helpful links on this.
UPDATE: @Ron-Maupin correctly points out, WAN is actually just a description of breadth of the network in the physical sense. Logically a WAN and LAN don't have to be any different. I would say, WAN is usually the upstream link to another administrative domain or a different network, but that doesn't have to be true. Actually, I haven't found the original author or source of the distinction between LAN, MAN, WAN and what not.
Actually, e.g. in Linux, routing is prevented by default (rp_filter setting, ip_forwarding=0) between "private" (Your LAN) addresses to prevent some bad behaviour by mistake. Generally, how You treat You addresses is up to You and in some cases the ISPs or other networks, You connect to.
How does the whole WAN addressing work?
First of all, the addresses are syntactically the same, we may assign them a different meaning.
So You might ask, how do other networks (or LANs) know, how to reach the other networks? Well, they could phone each other or write a letter and determine, who has what addresses. That is how the internet basically worked at the beginning and how some networks communicate with their provider (ISP). Some very important configurations in the internet are probably done in a similar way too. Most networks actually communicate this information by using a dynamic routing protocol. For participation, You apply at IANA or Your RIR for an Autonomous System Number. To obtain one, You have to give at least two other ASNs - Your upstream (WAN) connections. This is to prevent a split of the network, if one connection has a problem. After You have paid Your membership fee (which is about 2000 € for the first year here in Europe), You will get an ASN assigned, You will also very likely get some public IP addresses, so You can actually connect to the other networks so You can setup the connection over which the dynamic protocol (BGPv4) works. How this gets configured can be seen in a video (Keith Barker explains stuff in a simple enough way).
What they don't tell You so much is, in Your test lab, You can use any number or address You want. It should work all the same - just be so kind and don't connect it to the internet until You follow the standards. Actually, Keith Barker uses public IPs, such as 126.96.36.199 and 188.8.131.52 often in his videos, because they are simple.
Well, in IPv6 there are no private IPs, but IPs with special use too. Keith Barker has good videos on how IPv6 works.
I am a hands on guy, I need to see it work to really understand -> Read on
All of this, You can actually try out with free software on one computer at home. For that, I would recommend You get comfortable with GNU/ Linux, such as the Ubuntu 18.04 LTS, which has great support for network namespaces (by David Mahler) that can be used to simulate many computers on one machine in a fast and efficient way. For BGP you can use the BIRD routing daemon.
David Bombal has some good training videos on networking also.
So in general, the addressing is more or less same and You don't need special equipment to be able to really participate on the internet. What You need is the knowledge, the AS number and the related money and work. That is it, no black magic!