There is a tool called screen. After you login to a server, just type screen and from now on your session will be running in a virtual terminal that screen provides, which means it will be "kept alive" and still running even if you close your terminal (but not if you type exit - this will terminate screen !). You can also purposefully disconnect from screen session by pressing Ctrl+A and then D while in screen session and the session will still be running. Then you can log out of the server.
As said in the comment, after reconnecting to the server type screen -dr and you should be reconnected to your session. If you had only one disconnected screen session, you don't need to type any arguments after screen -dr. Otherwise, you can list all screen sessions on your account with screen -ls and then type screen -dr PID where PID is the PID of the session you want.
It's not quite "saving" a session as you requested, because the session is running all the time, even after you disconnect, which means if you left some program running, it will continue to run and continue to write output - if any - to the virtual terminal. screen has a scrollback buffer, so if you press Ctrl+A and then Esc while in screen session, you can scroll back to see the lines of output the program produced while you were disconnected. But if there is lot of output, the scrollback buffer may be not enough.
screen can do much more - for example you can have multiple sessions running in background and switch between them, but the basic use case described above should be enough for what you need.
