Where should I move it?
/opt/
That is the directory for 3rd party applications (ie. tar.gz installations). Mind that you can also use the debian installer if you downloaded those using dpkg but that also means you need to download and install dependencies.
tar.gzs have all permissions already correctly set so it is best to do
sudo cp {downloadedfile}.tar.gz /opt/
cd /opt/
sudo tar xvfz {downloadedfile}.tar.gz .
then cd into the new directory and execute the main program.
What do I need to do to run it by the terminal?
See inside the directory as it depends on what you install. Generally there will be a "readme" explaining what to do (sometimes you need to copy a file, like a serice, into the system or change an environment variable and your PATH so it is available to the system) but there will be an executable in there too that is the main program to start what is installed.
How do I get the shortcut in the apps overview?
Check for a .desktop file inside the directory and copy that over to
/usr/share/applications for universal and admin usage
~/.local/share/applications for personal usage (when there are more than 1 user you can copy it to all those homes).
If there is none check one of those two directories and copy one to a new name for that application (the name tends to be {application}.desktop) and edit the contents to point to the directory and executable in /opt/.