Run Bash script in background and exit terminal

To avoid exit signals propagating to child processes of the terminal and shell, run the command with nohup, i.e.:

nohup cmd &

Using screen:

screen -S <session name> -d -m <your command>

after that you can quit the terminal, also you can reattach to it by:

screen -r <session name>

More info: reference

If you want to run a specific command or file every second or so in the background after exiting the terminal you could try this easy little thing;

nohup watch -n5 'bash script.sh' &

That would run scipt.sh every 5 seconds.