meshcore is a system whose purpose is to send short 150-character messages without relying on the internet, telephone infrastructure, post pigeons, etc.
We have the internet, in which I can send gigabytes of data. We have SMS, and my grandpa has pigeons. Why do I care about another infrastructure and tool?
In case of an apocalypse, blackout, or "atomic winter" all the methods you mention will fail. In fact, meshcore (not only in theory) could survive for some time if suddenly the sun goes out (or disappears - i.e., NASA decides to shut it down).
On a more serious note, and in more real-life scenarios, such a thing could be helpful if your gov decides that, for your own safety, it shuts down the normal infrastructure (okay, maybe not pigeons) because of "safety reasons."
https://www.computerra.ru/339835/vnedrit-li-telegram-mesh-set/
Another example is natural disasters such as floods, fires, or hurricanes, which can undermine normal infrastructure. Similar to meshcore, the system (ClusterDuck Protocol) was successfully tested in 2020 during an earthquake in Puerto Rico. it werked.
https://www.programmingelectronics.com/clusterduck/
Ok, let's suppose that maybe you live in a more peaceful place. meshcore could be very useful when hiking, where mobile coverage is limited or unavailable. You might be hiking with a group and sharing messages:
https://www.youtube.com/watch?v=WRSfXiuQyc4
Last but not least, it's just fun. Testing how far you can send a message, participating in building infrastructure, or taking part in local crisis communication drills can be an interesting way to spend free time and meet new people.
At least for some autists.
Before buying anything, look up if there is any existing infrastructure of "meshcore nodes" in your local area. Nodes are devices, often solar-powered, whose purpose is to forward messages between other nodes and from/to regular meshcore devices (called companions).
You can check that on https://map.meshcore.io or other similar sites (search by the phrase "meshcore map").
As you can see above, Bratislava in Slovakia has a very rich network of nodes.
The second thing you need is a "meshcore companion". It is a small device with a chip, antenna, and battery, in a plastic case. Make sure it has a battery and the chip is somewhat battery-saving (it lasts at least a few days after charging).
Just go to your local classifieds or amazon, ebay, and search for "meshcore" or "meshcore companion."

A lot of enthusiasts sell very cheap used or new devices, sometimes even as low as $10 or $20. Sometimes in the description, both meshcore and meshstatic are mentioned. If you don't want to deal with flashing the device, ask the seller which of them is flashed, or ask them to flash it for you.
If you're still not sure if what you're buying is "good" a better option could be to make contact on some discord chat, Facebook group (God forgive me), or even visit a local ham radio club for advice. From my own experience, people in that community are extremely helpful. Don't be afraid.
Now we need an app. Unfortunately, the desktop meshcore apps on Linux are pretty bad. They are bloated, electron-based, and very heavy on older computers. Terminal-based apps are much better, but I assume terminal people wouldn't need a tutorial like this.
If you're sure you want a desktop app, the best one is the web app at https://app.meshcore.nz.
If you are using a Chromium-based browser, before entering this site, enable Experimental Web Platform features in chrome://flags/#enable-experimental-web-platform-features and restart your browser.
This app looks the same as the smartphone ones and is also available for Windows and macOS. Links to smarpthone apps and others below:
Ok, so make sure your companion is turned on (most companions have two buttons, left and right, or top and bottom). Turning it on is as simple as pressing the left (or top) button.
After turning it on, you should see something like:
PIN: 123456
This PIN is needed to authenticate that you are connecting to your device.
Now it's time to run the meshcore app on your device. Select Bluetooth (or BLE) from the connecting methods, and then "connect". You should see a field to enter the PIN. Enter it.
Now, let the world know you're here. In the top right corner of the app, there are three icons. Select the first one from the left, then click "Advert to all".
After a few moments (one minute max), your contact list should start filling up with various devices: companions, repeaters (aka nodes), and room servers.
That's pretty much all. The app is very simple. Adding contacts usually involves searching for them in your contact list (these ~300 contacts that are already there are simple and close to you) or by adding them by their public key.

Most of the life happens in so-called hashtag channels. To access the channels page, at the bottom of the app, we have 3 options. Just select "Channels". The "Public" one is added by default and available right away.
Useful channels to add are #CITY (replace CITY with the name of the city you're in), #test, #bot. It's also good to add other cities within a 200 km radius.
Adding channels is simple too. Click on the 3 dots in the top right corner, select "Add channel," then "Join hashtag channel" type its name (as suggested above), and confirm.
When you send a message, it's important to check whether it was "Delivered" or "Received" (in the case of channels). For 1:1 messages, the app sometimes tries up to 5 times if sending fails, and for channels, if no node "hears" it after a while, the message will remain only on your device.
I didn't mention this before, but meshcore operates on very low power. That's its strength and weakness. The more nodes that are able to transform your message, the better the chance it will be delivered.