Digital noise is caused by a combination of factors... pixel density on the sensor, high ISO "gain", and lighting conditions. ISO 3200 is pushing it with your camera. For that matter, ISO 1600 can be fairly noisy, too. Lower than that is no problem, so long as you don't underexpose.
I have not used a T5i.. but I have used another camera with the same sensor and a generation older processor... the original 7D. Your camera has an 18MP sensor and a Digic 5 processor to handle the images. My two 7D are also 18MP cameras and they use dual Digic 4 (primarily to support a faster frame rate and AF system that was advanced for its day).
I almost exclusively shoot RAW images, which means no noise reduction was being applied in the camera. While I would use ISO 1600 and 3200 when needed, sometimes even ISO 6400, I would always need to use some noise reduction to finish images. The NR software I used was Imagenomic Noiseware, as a Photoshop plug-in.
It is also critical to avoid underexposure. If your images are too dark and need to be brightened up in post processing, that will amplfy the noise a lot. As Trevor suggested, one thing that can help with this is Expose To The Right (ETTR), which basically means dial in some + Exposure Compensation when using any of the auto exposure modes or slightly overexpose fully manual settings. In Av, Tv or P (rarely used) I would dial in +2/3 stop for ISO 1600 and +1 to +1.33 stops of E.C. with ISO 3200, With fully manual settings, I'd try to overexpose by about 1/3 stop, but a wee bit more was okay. I always found it better to "pull back" images from being too bright, than to "push" image that were too dark. With these settings I would even use higher ISO 6400 if needed (same settings as ISO 3200).
I learned to use ETTR with older Canon DSLRs. I think it would help you too, The original 7D were the last model I had to do very much of it. I now shoot with 7D Mark II and find they don't need very much of this kind of tweaking.
Oh, and don't be silly when reviewing and evaluating your images on your computer monitor. An images from an 18MP camera "at 100%" on the typical monitor will be like making a 50 inch wide print, then viewing it from 18 or 20 inches away. That's something you'd never actually do. If you made a print over four feet wide, you'd stand a lot farther back to view it.
While it's okay to zoom in to 100% or even higher to do careful retouching to images. But the noise you see.... among other things... will disappear when you view the image at more normal sizes. Back off to 25 or 33%... If you don't see noise then, you sure won't see it in a print that size or in a reduced resolution image to be posted online.
Finally, when post-processing images, always do noise reduction before doing any sharpening. In fact, sharpening is usually best left to one of the last steps, after any resizing has been done.
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Alan Myers
San Jose, Calif., USA
"Walk softly and carry a big lens."
GEAR: 5DII, 7DII (x2), 7D(x2) some other cameras, various lenses & accessories
FLICKR & ZENFOLIO
