I can also attest to the 18-55 and 55-250 EF-S lenses being really quite good optically.
However, and this is quite important for me - the vast majority of my shots fall in the 30-100mm or so range... which no one single lens of the two provides, and I was forever swapping lens back and forth.
I recently plucked up some big wallet based cojones and stumped up for a 24-105 L f4.0 IS USM. This decision was partly because some lens rentals for a holiday to south africa got me addicted to the build and image quality of the L lenses and partly because I operate on a buy once type policy as much as possible... I could have bought an intermediate step lens but I would have ended up replacing it in the not so distant future because I know what I'm like!
I have to say that having used it a bit now I love the 24-105mm range, it is so useful for my purposes and I can happily take out the camera with just that one lens most of the time. Of course 18-135mm is an even more extensive zoom range and will suit an even wider range of photographic opportunities.
So I guess what I am saying is that, expensive L lenses aside, you sometimes have to sacrifice a bit for convenience. Even with L lenses you sacrifice convenience for quality as the L series primes (fixed focal length) have the highest quality of all whereas the L series zooms are a compromise for flexibility, albeit a compromise from a higher starting point.
As mentioned the build quality will also be much improved - the kit lens is a bit toy like after you've played with more expensive setups. Chances are you will find the image quality absolutely fine as well... plus things like barrel distortion and pincushioning which is where the 18-135 will be most likely to fall short of the 18-55 and 55-250 in terms of Image quality are easy to correct in photoshop when they are visible, and most of the time they probably won't be to the average chap like you and me!
If it were me, I'd go for the 18-135 over the 18-55... then further down the line pick up the 70-300 which is a great lens for the money by all accounts. The idea of having one lens pick up where another stops off is great and all, unless that focal length change over happens to occur slap bang in the middle of your most used ranges.
Different people, different shooting styles, different answers I'm afraid!