Finding a kit to shoot "everything" is not easy. You'll end up with way too many lenses.
I noticed that you've mentioned "travel" several times and it's also in your signature. If travel is the first of your "everything" and you want to travel light, the 17-55/2.8 and the 55-250/4.5-5.6 make a good combo. I travel with the 17-55 and the 70-300/DO, but this combo is expensive and heavier. From what I can see and read, the 55-250 seems to have good IQ.
About not wanting to buy an EF-S lens, been there and done that>>for about 6 months. But when I thought about what I needed for a trip to Monterrey, CA, I decided I needed the f/2.8 and IS for museums, night shots and the aquarium. The second best purchase I've made so far (i.e., it's the second most used lens in my kit).
Yeah, traveling is important to me and I think the limitations of being on the road somewhere forces you to choose certain lenses. I don't shoot in a studio and I don't even shoot a lot indoors, so my lens choices reflect that.
I follow the "travel rule" of not traveling with more than you can carry by yourself.
I just bought the EF 17-40, which I will use along with the EF 70-200 F4 IS. Next up is probably the 24-70 or the 24-105 (replacing my 28-135).