You don't absolutely need triggers for your YN-468 and new YN-568 to work together. The both include built-in optical triggers that support both ETTL and manual exposure control. The YN-568 will work as an ETTL MASTER, so it will control and set exposure automatically on the YN-468. In manual mode, either model can be a master or slave.
What external triggers do is substitute radio frequencies (RF) for optical triggering (detecting the light from another flash). Optical triggers can be unreliable if they can't "see" the light put out by the triggering flash. RF improves reliability because it is not (easily) blocked by obstacles such as reflectors, and don't have to be positioned to see the triggering flash's light output.
All an RF-603 will do is give you wireless RF triggering. The RF-603 does not support ETTL, so the flashes would need to be setup for manual exposure. The Yongnou YN-622C triggers will support wireless RF triggering AND ETTL, so the flashes could be setup for automatic or manual exposure control.
I think that as you get more experience in multi-flash setups you will find that manual power control is the way to go. Both your YN-468 and YN-568 can be setup for manual power settings. But consider you can buy two YN-560IV flashes, which include built-in RF triggers, for the price of one YN-568. Then you could buy a YN-560TX transmitter to trigger and control the power levels on both flashes remotely from your camera.