if I run a job in background here I still see the output in the terminal...
you can store the output in a file and check that file later like
./bigjob.sh > output.txt &
but I guess you know that.
A running job can be stopped with CTRL+z. If you type
it continues execution in background. With
you can reattach it if you want.
Have you tried screen already?
With this tool you can start the big job and even close the terminal (or close the connection if it is a remote host) and later reattach it.
This would work something like that
screen -d -m bigjob.sh
screen -d -r
with CTRL+a CTRL+d you can detach again.
Perhaps it helps...