I have a Canon T5i and want to get 2 of the YN 560 ii speedlites. Do I need to get the TN 560 flash controller or can I get the 603c transmitters?
The YN-560II flash is manual only and has no radio triggers built in, so you must add a radio receiver to each light, and a transmitter to trip the lights. RF-603II units are transceivers and can be both transmitters or receivers. The dedicated YN-560-TX transmitter makes more sense if you are using YN-560III or YN-560IV flashes which have a radio receiver and remote power control built in; or RF-605 triggers that allow for group control.
However. The YN-560II also has optical slave modes S1 & S2, which can be tripped by the pop-up flash of your T5i. Do not confuse this with Canon's wireless eTTL system. The pop-up flash needs to be OUT of wireless mode. And if it's in M mode, you can use S1 on the YN-560II; if the popup is in eTTL mode, you can use S2 on the YN-560IIs. You will need "line of sight" (where the red panel on the front of the flash has visibility to the pop-up flash of your camera; the pop-up flash has to fire; you have no remote power, eTTL, or HSS capability; and this scheme works less well outside without bounce surfaces and a lot of ambient light.
If they are all the yongnuo brand, do I even need the 560 Speedlites to have the 603 attached or just one that sits on the camera?
If you're using the RF-603 triggers, you can use them with any ISO-compatible hotshoe flashes (i.e., any hotshoe flash that fires from the center pin). They don't need to be Yongnuo flashes, but Yongnuo flashes are among the cheapest out there, which is probably why you picked them in the first place.
This is for a studio set up so I just need to know what I need with two SpeedLights so I don't spend money on the wrong product.
Actually, you first should probably figure out if you can do what you need to do with only speedlights, or if you might need a bare bulb flash, or a studio strobe instead. Speedlights are the small, light, convenient choice, but are underpowered compared to other types of flashes. And even if you know you want a speedlight, you also need to evaluate whether you want that flash to do double duty as an on-camera run'n'gun flash for event/social shooting. If so, then a cheap manual-only flash may not be your best choice, and an eTTL capable flash, such as a YN-568EX or used 430EXII, might be a better choice. (See: What features should one look for when selecting a flash?).
To put together a basic Strobist set up, nearly any kind of flash radio triggers will work, but even more importantly, you will need stands, umbrella adapters (to attach the speedlight to the stands), and probably some kind of modifier (umbrellas are a good low-cost starting point), and possibly a bag to put it all in. As with camera bodies and lenses, there's more to it than just buying speedlights and triggers.