I purposefully skipped the 70-200mm f4L non IS.. It's a great lens, but I wouldn't recommend a non-IS telephoto lens as a first telephoto.. I guess it depends how comfortable you are shooting without image stabilization, because at telephoto focal lengths, non-IS is very very obvious.. You'll need to pay attention to your ISO settings and your f-stop a whole lot more. The 600D is capable of giving you good results, you're just going to have to work with your camera a lot more.
Here's the difference between the 70-200 non IS and the 55-250 IS, shooting-wise.. We'll use 200mm as a working focal length for comparison.. (As in, the 55-250, we're only zooming in to 200mm..)
The rule of thumb is, without image stabilization, you can take a hand-held crisp image with a max shutter speed of 1/focal_length.. so in our case, that will be 1/200 of a second.
70-200 non IS: up to 1/200 of a second.. no IS, means no better results. This is the lens's limitation..
55-250 IS: *with* IS, you get "4 extra stops".. let's be conservative and say you get 3 stops.. that means we take our 1/200 and half-it 3 times.. 1/200 -> 1/100 (one) -> 1/50 (two) -> 1/25 (three). So, the 55-250 IS will let you take pictures at 200mm at shutter speeds all the way to 1/25 of a second. This is a HUGE advantage over the 70-200 non IS.
*However* shutter speed is only 1 part of the picture-taking equation.. You've got shutter speed, f-stop, and ISO.
shutter speed advantage: Already covered.. the 55-250 IS has a 3 stop advantage over the 70-200 non IS.
f/stop advantage: The 70-200 non IS is a constant f/4 (f-stop), while the 55-250 is f/5.6 at 200mm.. f/4 is 1-stop brighter than f/5.6, so the 70-200 non IS has a 1 stop advantage over the 55-250.
We can't lower the shutter speed on the 70-200 non IS if we need more light, but we CAN open the aperture (same as f-stop) more than the 55-250...
So technically, the 55-250 has a 3 stop shutter speed advantage, but a -1 f/stop advantage.. So it has in total, a 2 stop "picture-taking" advantage over the 70-200mm, if that makes sense..
Example: a sunny day is f/8, ISO 100, 1/250th. Both lenses can just shoot away on a sunny day, no problem.. But a cloudy day might be f/8, ISO 100, 1/60th..
Without even thinking about it, you can just shoot away with your 55-250 IS no problem.. It can handle 1/60 of a second.
With your 70-200 non IS, you CAN'T just shoot away... You need to drop f/8 to f/5.6 to get you from 1/60th to 1/125. That still isn't fast enough of a shutter speed, so you need to drop f/5.6 to f/4, which gets your shutter speed from 1/125 to 1/250th. OK, now you can take hand-held crisp images.
What about in the late afternoon..? Let's say you've got a reading of f/5.6 and 1/30th..
With the 55-250, you're GOOD.. it can handle 1/30th of a second.. Point and shoot.
With the 70-200 non IS, again, you gotta make some changes.. drop f/5.6 to f/4 and your shutter speed goes from 1/30th to 1/60th. That's not enough. Now you need to up your ISO. ISO 100 to 200 takes your shutter speed from 1/60th to 1/125th. Still not enough. Now you need to bring your ISO from 200 to 400, which takes your shutter speed from 1/125th to 1/250th. NOW you're good.
What about later in the evening, when it still doesn't even look all that dark out, but it actually really is..? f/5.6 and 1/30th and ISO 800..? (depending on lighting, this is can be darker or brighter than indoors..)
With the 55-250 you're GOOD (though you had to change your ISO settings to get there...)
With the 70-200, you drop f/5.6 to f/4 which gets you from 1/30th to 1/60th... Now you need to up your ISO again to 1600, which gets you from 1/60th to 1/125th.. Again you need to up your ISO from 1600 to 3200.. which gets you from 1/125th to 1/250th. And you're good. Except you're at ISO 3200.. Which is a compromise.. Your camera can handle ISO 3200 if you're not making huge prints, but it sure isn't ISO 800.
As for ISO, it's all dependent on your camera.. And YOUR camera has excellent ISO options.. Let's say ISO 100 - ISO 800 are all considered equal (or close enough) on your camera. About in the ISO 800-1600 range is when you start losing a bit of contrast.. Perhaps about the contrast that the 70-200 non IS has over the 55-250.. It's a toss up.
Remember, that's hand-held.. If you're putting your camera on a tripod, then the 55-250 no longer has that shutter-speed advantage. In fact, the 70-200 now has the 1-stop f/stop advantage! Also, if you're taking pictures of people, 1/25th is a bit slow.. I wouldn't trust people to stay still much better than 1/50th or so.. If you're taking pictures of animals, it all depends on the animal.. You might need to approach 1/200th of a second regardless, in which case the 55-250 loses its advantage..
In all seriousness, those examples are real situations (and real available-light situations), but they're also over-simplified.. If you can steady yourself with that 70-200, you're going to get better shutter speeds than 1/200th. And if you're actually carrying around a tripod, it's all a wash. If you're doing flash photography, it's all a wash (or at least less cut and dry.. and 200mm is generally not flash territory.). As you get into "your-way" of photography, all these things sort of come together and you know what YOUR limitations and photography needs are, and you know if you can consistently take crisp images with a non-IS telephoto.
But as an all-around-I'm-just-getting-into-telephoto-territory, I think it's rather frustrating to get a non-IS lens. Once you know how you use it, maybe the IS is no longer important.. But I personally think IS is the better way to start.