Joining a public Minecraft server is very very easy. Just open the Minecraft launcher and click Play to run the main Minecraft game. Once you reach the main Minecraft title screen, you click the Multiplayer button, then click the Add Minecraft Server button to add the Minecraft server.
Now type a name for the Minecraft server in the Server Name box, then type the Minecraft server address in the Server Address box. Typically this will be a domain name, or an IP address, such as 184.108.40.206. Then click "Done" to add the Minecraft server to your Minecraft server list.
Then it's simply a matter of clicking a Minecraft server in the list, and clicking the Join Server button to connect to it. Now let's look at a couple of issues that may come up when joining a Minecraft server.
Whitelisted Minecraft servers are protected by a whitelist — that is, a list of usernames that are allowed to join the Minecraft server. To join a whitelisted Minecraft server, you need to apply to have your Minecraft username added to the whitelist (a list of minecraft names that are able to join). Typically, this involves filling out a form and waiting a few hours or days. Most Minecraft servers require you to put in separate applications for yourself and for your kid. Once you're on the whitelist, you join the Minecraft server as described above.
Non-whitelisted Minecraft servers do not have a whitelist, which means anyone can join the Minecraft server simply by entering the server's address in their Minecraft client, as shown above.
Applying for a whitelisted Minecraft server can be a bit of a drag — especially when you have an impatient eight-year-old tugging at your sleeve — but it does provide an extra degree of reassurance that all the Minecraft players on the Minecraft server are known to the server administrators.
One thing to watch out for when joining a Minecraft server is making sure the version number of your Minecraft client matches the version number of the Minecraft server. Generally speaking, the Minecraft versions have to match, or you won't be able to connect.
For example, the current version of Minecraft is 1.16.4, but many Minecraft servers are still running 1.12.2, or even 1.9.4. Sometimes the Minecraft server's website tells you what version they're running, but sometimes the only way to tell is to try it and see.
Fortunately it's easy to switch to different Minecraft versions of the Minecraft client. In the Minecraft launcher, you can click New Profile to create different profiles that use any Minecraft version you like. Then, just select the Minecraft profile you want to use from the drop-down list in the Minecraft launcher, and click Play.