I use RF-602's.. I have two sets, one for the shutter, one for the speedlites. Sounds like that may not be an issue now though, since the 603's are RX+TX in one.
If you want to dedicate a trigger to each flash, then yes, you need one to act as transmitter (Tx) and 3 to recceive (Rx), one for each flash.
The camera will send the flash signal to the hotshoe when you press the shutter button. Whatever is in the hot shoe will get triggered, if it's a flash you get a flash.. if it's a Tx, that will send a signal out to the Rx units. You can do both if you want as well, sit a flash on top of the trigger, sitting in the hot shoe.
Keep in mind though, that the YN-560's have optical slaves.. so if you plan on using these indoors and within line of sight from the other flashes, you don't need a dedicated trigger for each unit. You could have the Tx on the camera send a signal to on Rx unit, and when the Rx unit triggers the flash to go off, the other two YN560's can be set in slave mode to "see" the flash, and then fire as a response. Of course if you start hiding flashes under / behind / around things, they may not see the original flash, and then they won't fire.. just depends on your needs.