The Yongnuo 565EX does everything the 580EXII does, except High Speed Sync and wireless optical master control. Add Yongnuo YN-622 radio transceivers and they'll do everything (except HSS) that a 600EX-RT will do. They are built from very nice materials. They do not have such a strong reputation for reliability as most Canon flashes but if that's a concern, just buy another one as a back-up. I've had no problems with mine.
I now have a 565EX, a lower powered YN-468 II and another 565EX on its way to me, along with four YN-622 E-TTL II transceivers. For all this, I paid about 60% of the price of one 600EX-RT. Canon vs third party flashes has been a hotly debated topic recently. Professionals will often say they need the dependability of the Canon brand, regardless of cost. I agree that if price is no object, there's no reason not to go with Canon. Otherwise, my view is that Yongnuo light is just as good as Canon light.
Edit: to answer your specific question about triggering an off-camera Yongnuo 565EX, it depends on whether you want it to expose automatically. If so, your best choice would be a pair of YN-622 transceivers. If you are happy to set the flash output manually, it can be triggered by your built-in flash or by any cheap radio triggers, e.g. RF-602 or 603.
Re-edit: the new Yongnuo 568EX adds High Speed Sync but doesn't have an input for an external battery pack.