If you plan to do this learning to computer lark then you will probably want to start reading guides with an eye to replicating it on another system, or with other programs. Learn what it is that is being done, not just steps to do something.
Back in the day you would have needed a crossover cable. You can make one from a normal ethernet cable if you want but today most network ports have a switch internally to do it.
Once you have that you have a million different methods.
You will probably want to make sure your computers have IP addresses they can talk to each other on. I don't know offhand is Linux Mint uses a fallback option so prepare to set static addresses for each. You could set up a DHCP server if you wanted. In my case I am lazy so I would go with static, or actually more likely just grab a router from the pile as it will already have nice DHCP for me to use and I have to do there is plug it in.
Most Windows guides will probably have you use file sharing. More technically this is known as SMB file shares and Linux (and everything else) has a top notch implementation of it via a program called SAMBA. It can be a pain to configure though (SMB actually handles a lot more than just file shares if you want it to but we will skip domains for now) so might want to grab a UI to do it (though if you are bored do learn the steps for a command line, or fish out the config text file after all is said and done and learn what each change did).
The million other ways refer to the million other ways. In my case I usually go back to FTP. It is old. It is complicated if you let it be. If is primitive in a lot of ways but when I have to transfer a few hundred thousands or so files (easily done if you are copying a hard drive for a fairly normal person's machine) it is a thing that works. FTP is server-client but communication is two way so you can either upload or download from them. If both machines are sitting next to each other (or the few steps the average LAN cable reaches) then who cares as you can shuffle things around on the machine with the client if you like. If not then set up servers on both and clients on both.
You might want a bit more security at some point so maybe look up sftp (note a different thing to ftps) but I will skip that one for now.
In the common stuff next is file transfer over SSH (see also SCP, which technically did lead to SFTP). SSH is also good for remote controlling a machine, typically via the command line but you could go through all the steps to enable you to start an x session over SSH.
While most will probably use SMB via SAMBA then Linux does have its own means of network file sharing known as NFS.
Rsync might be something to look into. It is mostly about synchronisation of files (think you have a document you edited and want to make sure the one on the network is now up to date, or that the one you are opening is recently pulled from the network and thus the most current version).
Webdav I will mention just because. Don't think I have ever used it myself but is another thing.
If it is just Minecraft you can set the server to not use internet looked up accounts and thus join whatever you like if you are sharing a pirated version or something.
If you want a static IP then it is much the same as any system. I changed from mint a while back but it was about what I would expect, and what I have now is similar enough. Go find the advanced network settings/connection settings (whether it is available on a simple click on the network icon I don't know) and in there will be something for it, possibly in IPv4 settings.
The crossover cable is just a simple rewire of a conventional cable. As mentioned earlier most modern NICs (network interface cards) have auto switching done in hardware/drivers so unless you are using exclusively old computers (none of which are likely potent enough for minecraft) nobody has likely "needed" a crossover cable in decades.