When it comes to image quality, in particular, all the Canon crop cameras (aside from the entry level T3 or 1100D) use essentially the same 18MP sensor and virtually the same processing as your 60D. So, don't expect there to be any noticeable difference.
The T4i/650D uses a newer processor, so might see very tiny bit of difference. But, aside from that, I'd expect it to be more of a downgrade or parallel move, than an upgrade from your 60D.
The 7D uses two processors to allow it to shoot faster... more frames per second.
For the large part, among the current Canon crop sensor cameras the differences are not related to image quality at all... The differences are in shooting speed, autofocus performance, control layout, build/durability/sealing and customizability, and other features... 7D would be the primary bump up from your 60D in terms of AF performance and shooting speed. If you shoot lots of video, the T4i/650D might be considered an upgrade.
You'd have to bump up to one of the full frame models to see any significant difference in image quality. Even then, unless you print big... really, really big!... you won't see much difference. The 5DIII or 5DII will give you a bit more resolution and cleaner high ISO images (one or two stops worth). Just don't expect for the IQ difference to be noticeable at anything less than about 16x20" or larger. Even then, it won't "blow you away". And, if you go full frame, you should expect to reconfigure your lens kit to accomodate full frame. At the tele end of things, you'll need bigger, heavier, much more expensive lenses to enjoy the same "reach" you have now with your crop camera. At the wide end, you would see less distortion in most cases because you can use less radical lenses... but FF wide lenses will be bigger, heavier and more expensive, too, in order to give you a large enough image circle to cover the sensor.
Seriously, your best bet would be to stick with the 60D and look to your lenses. That's where you're lacking. Yours are mostly the more entry-level lenses, which are adequate, but lightly built and not as capable of top IQ and fast focus or don't have as good IS as some of the better lenses in the Canon line-up. For example, a 70-200/4 IS or f2.8 IS II would be a significant step up from your 55-250. All the Canon 70-200s are pro-quality work horses. If you need more reach than 200mm, a 300/4 IS or 400/5.6 might be a good, affordable choice... Or there are some super telephotos if your budget is quite big.
A 17-55/2.8 IS or 15-85 IS would be a big improvement over your 18-55 or 18-135. The 17-55 gives you f2.8, while the 15-85 is more of a compact, versatile walk-around lens with f3.5-5.6 aperture.
Your 50/1.4 and 85/1.8 are perfectly capable, used right. There are more expensive uber lenses available, such as the Canon 50/1.2L or 85/1.2L II... but those are big increase in prices. Sigma 50/1.4 and 85/1.4 might be worth considering, but are bigger and heavier and more expensive than what you have now.
I'd want something wider than 17mm, too... (the 15-85mm might be wide enough for some folks). Personally I use a Tokina 12-24/4, but the Canon 10-22 is another excellent choice. There are other ultrawides, at various price points.
Build the right lens kit and you may some day want to augment your 60D (or a 7D) with a FF camera, such as the 6D, so that you have the best of both worlds.
One feature the 60D lacks is Micro Adjust. That allows you to fine tune the focus accuracy of your lenses. It's not so critical with smaller aperture lenses (f4, etc.), but might be more important with big apertures (f1.4, etc.) and shallow depth of field effects. 7D has Micro Adjust. So did the 50D (though that now-discontinued model is slightly lower 15MP resolution, without video, not quite as high ISO capable and lacks some of the 60D's features... some still like it better and it's IQ is darned close to 60D).
So, I'd suggest you not worry about the camera too much. Put your money into lenses instead. You'll see much more improvement. Lenses are always more important than the camera they are used upon. Lenses determine how the image will look in many ways, while the camera just captures what the lens "sees".
To get more "pop" out of your images... get the matching lens hoods for all your lenses. Get a top quality circular polarizing filter or two, to fit key lenses in your kit. Don't use "protection" filters unless really, really necessary. Especially never use cheap, single or uncoated filters of any type. Only top quality, multi-coated... and only when they can actually enhance your images.
Get out and shoot more, practice more. Also work on your post-processing... that's often overlooked and has much room for improvement. A better quality computer monitor and a calibration suite for your computer might make for some major improvements to your images.