I understand that zoom lenses "reach" further on crop sensor cameras. Does a zoom lens specifically designed for a crop sensor camera have that effect? For instance, does a 55-250 mm zoom made for a crop have an effective reach of around 400 mm? Or does that only apply to lenses that work on both full frame and crop?
250mm lens on a crop sensor camera will give exactly the same angle of view regardless... doesn't matter if it's a "crop only" lens or if it's a "full frame capable" lens.
If you have only ever used a crop sensor camera, forget about "effective reach". It won't mean anything to you. The only time you have to concern yourself about the "effective reach" is when going back and forth between formats. For example, if you used to shoot with a typical 35mm film camera & lenses and are now trying to buy lenses for an APS-C crop sensor camera that will perform similarly on it, you'll need to do some math to help with lens selection. Or, if you have been using a crop sensor camera for a while and decide to switch to full frame or to add a full frame camera to your kit that will be used alongside your crop camera, you'll need to do some math to know how any compatible lenses you currently have will "behave" on the larger format and what lenses you might need to add, depending upon your purposes with the camera.
Depending upon what format it's used upon, any given focal length will "behave" differently. Just for example, on an APS-C camera a 30mm lens acts as a "standard" lens... not wide, nor telephoto. But on full frame it would be a wide angle lens. For that matter, on a medium format camera with an even larger image area, a 30mm lens would act as an "ultra wide". Or on a digital camera with a tiny 1/2.3" sensor, much smaller than APS-C, that same 30mm would behave as a telephoto!
All you really need to know, though, is how focal lengths "behave" on the camera format you use. You only need to worry about "conversion factors" when you use multiple formats of cameras or are switching between them.
It sounds as if you are using a Canon 70D, which uses an APS-C sensor (1.6X "lens factor"). On that camera, you'll find the following:
10-22mm is an ultrawide to wide angle zoom
17-55mm or 18-55mm or 15-85mm or 18-135mm is a "standard" or "walk-around" zoom that covers wide angle to standard to short telephoto or moderate telephoto.
55-250mm or 70-200mm or 70-300mm or 100-400mm are more powerful short or moderate telephoto to strong or super telephoto zooms.
Another difference is that in many cases a crop sensor camera such as yours can make full use of both crop sensor and full frame design lenses. In contrast, a full frame camera basically requires full frame capable lenses. (Note: Some cameras allow crop design lenses to be used, but in most cases the image will be heavily cropped so it's sort of a waste of the full frame camera to use it that way.)
And, finally, lenses designed for crop cameras are typically smaller, lighter and less expensive than those designed for full frame. For example, I often use a $1300, 3 lb. 300mm f/4 lens hand held on a crop sensor camera to shoot sporting events. To have the same "reach" and speed (lens aperture) with a full frame camera I'd need to fit it with a $9000, 8 lb. 500mm f/4 lens that pretty much requires a sturdy tripod.