Moving to a new server — when upgrading specs or changing the operating system or network

The specs, operating system and network (zone) of a server you're using can't be changed later. If you need higher specs or want to change the operating system or network, you need to get a new server and move your data. If you follow the steps in order, you can move with almost no interruption to your service.

Overall steps

Order the new server — leave the old server as it is until the move is finished. If its expiry date is close, renew it first.
Move your data
Check that everything works on the new server
Change the address people connect to (update your domain, or share the new IP for a game server)
Wrap up the old server

Linux → Linux: copy directly between servers (rsync)

Copying directly from server to server, without going through your PC, is fast. Install rsync on both servers.

sudo apt install -y rsync

Connect to the new server and pull the folder from the old server. Replace username and old-server-ip with the old server's connection info. (It asks for the old server's password.)

rsync -avz --progress username@old-server-ip:/home/username/myfolder/ ~/myfolder/

Write the / at the end of the folder paths exactly as shown so that what's "inside" the folder gets moved.
If you've set the old server to log in with SSH keys only, it won't connect with a password. In that case, download to your PC and then upload to the new server. (Uploading and downloading files)
If it gets cut off partway, running the same command again continues the copy.

Other things to move

DB — Create a backup file on the old server, bring it to the new server, and restore it.

# On the old server
sudo mysqldump --all-databases > ~/db-all.sql

# On the new server (install the DB first)
rsync -avz username@old-server-ip:/home/username/db-all.sql ~/
sudo mysql < ~/db-all.sql

Configuration files like .env — If you moved the whole app folder, they come along with it. They're hidden files, so check with ls -a.
Scheduled jobs — View the list with crontab -l on the old server, and add them again with crontab -e on the new server.
Nginx configuration — Move your configuration files from /etc/nginx/sites-available/. Rather than moving HTTPS certificates, it's simpler to point your domain at the new server and then get new ones issued.
How programs are run — Register programs you ran with pm2 or systemd again on the new server, the same way. (Deploying your own web app to a server)

Minecraft server

Move it after stopping the server completely with stop in the old server's console. Copying while it's running can corrupt the world.
Move the entire server folder. (Including world, plugins, server.properties and everything else)

rsync -avz --progress username@old-server-ip:/home/username/minecraft/ ~/minecraft/

Install Java on the new server and start it. (Creating a Minecraft server (Java Edition, Paper and vanilla))

Windows server

The simplest way is to compress the folders you're moving (right-click → Compress), download them to your PC, then upload them to the new server. (Uploading and downloading files)
For a game server, stop the program on the old server before compressing.

Changing the address people connect to

If you use a domain, change the A record to the new server's IP on your domain management site. It can take anywhere from a few minutes to a few hours for the new address to spread. Then get an HTTPS certificate issued on the new server. (Putting up a website)
For a game server people join by IP, tell your friends or players the new IP.

Checklist before wrapping up the old server

Have you confirmed that your service works properly on the new server?
Are there any files left only on the old server? As a final step, make one more backup.
If you don't renew the old server, it will be stopped and terminated after its expiry date, and a terminated server's data can't be recovered.