sudo usermod -a -G groupname username
It's really simple like above, the -a is for append so that you are not changing their main group, but adding them to another additonal group. Just change "groupname" to your group and "username" to the user you want to be added to "groupname".
A common task these days is getting your user access to kvm for virtualization so the KVM/QEMU process doesn't have to run as root.
An example of adding a user to the kvm group:
sudo usermod -a -G kvm username
For KVM you'd also have to add access to /dev/kvm:
chown root.kvm /dev/kvm