I don't think full frame is worth it unless you know you need it (can't get images you want with your current gear) or unless you have plenty of money to spare.
I've had a T3i, 18-55 IS II, and 55-250 IS since 2012, and in the last year I've taken some incredible photos with them. I'm consistently amazed by how good the RAW files look. On a recent trip I took one photo of a flower that I couldn't improve with Lightroom; it looked best with no develop settings at all! The color is so purple that it's out of gamut for my monitor and the print lab I use. I have some photos from this setup that I think are worthy of printing relatively large and hanging on the wall.
Other lenses I have that I rarely use are: 50mm f/1.8 II, Tokina 11-16 f/2.8, Rokinon 35mm f/1.4
I've been wanting to try full frame for a while, and prices seem good right now and I can afford it. So I just bought a 6D, a 24-70 ii, and a Rokinon 14mm f/2.8. I haven't used them much yet. I'll probably buy a 100-400 ii soon. The two main things I will be able to do better with full frame are night sky photos and better blurred backgrounds (subject-background separation).
There are downsides to full frame. The larger sensor demands heavier, more expensive glass. I'm often hiking or backpacking, and I don't look forward to carrying several times the weight in camera gear. Quality wise, based on my interpretation of The Digital Picture's test charts, the $600 24-105 L has similar performance to the ~$100 18-55 kit lens. Yeah, the 24-105 is slightly better, but it's not six times better. And the 55-250 is similar to the larger more expensive 70-300mm f/4-5.6 IS USM. To me it seems like if you want a step up from the EF-S kit lenses in quality you need to spend thousands on full frame zoom lenses. If you simply switch to full frame and get the cheapest full frame zoom lenses, I'm not sure what the point would be.
The 24-70 ii is nice without many flaws in image quality, but I miss having IS, and despite popular opinion it's not a prime killer. Primes still have much wider apertures available, some of them are sharper when stopped down than the 24-70 ii, and some of them have IS. But primes are less convenient of course.
I would recommend sticking with APS-C and upgrading lenses. If prime lenses fit your shooting style those are the best value to improve your photography in my opinion. You don't need L primes, and you can get great performance from less expensive ones like the 35mm f/2 IS, 85mm f/1.8 USM. Some of the third party brands make excellent primes, and Rokinon/Samyang/Bower lenses are an incredible deal if you get a good one and if you don't mind manual focus.
If primes don't work for your style, there are some good third party zooms, too. The Tokina 11-16 is a great lens, though I don't use it very much because it's hard to take good wide angle shots. And I've heard good things about the Sigma 18-35 f/1.8 which is will get you full frame look and outstanding image quality on a crop sensor.
Even the cheap primes are not bad. I was going through my photos the other day, and I came across some nice photos taken with the 50mm f/1.8. Sure they probably aren't great if I zoom in and study the pixels (but I didn't do that), and the contrast could have been slightly better if I had a better quality lens, but they are not bad. I should use that lens more often.
You could consider upgrading your camera body if yours doesn't meet your needs in some way, such as not fast enough continuous shooting, or not a big enough viewfider (the 60D and 70D have larger, brighter pentaprism viewfinders, but I'm not sure if that's worth upgrading for). Some of the best features of the 6D relative to my T3i are the locking mode dial, larger buffer for RAW burst shooting (16 shots at 4.5 fps instead of 7 shots at 3.7fps), more buttons for faster adjustment, etc. All of those things are available in crop sensor bodies, too.
My APS-C kit cost about $1100 for a body and zoom lenses covering 11-250mm.
A full frame setup will be over $5000 to cover about the same equivalent range with significantly better image quality. But my images won't be 5x better, and there was nothing really wrong with my images before. I'm still not sure that I made the right choice.
I'm considering getting some more primes to go with my 6D (a 35mm with autofocus, 85mm, and maybe something longer. Or maybe I'll upgrade the 50mm f/1.8 to something that's worth using frequently), but I'm hesitant to give up the zoom lenses because they are so convenient sometimes. As a result I'll probably spend even more on lenses.
I do all kinds of photography. I like anything natural, so I've done a lot of landscapes and wildflowers recently. I like wildlife when it's convenient, so I need a good super telephoto. I'll do portraits when people want them. And I like to bring a camera with me when I hike or go on vacation for snapshots and in case I come across a good photo opportunity.