The 15-85 is a very good all round lens (based on the reviews I have read, I haven't tried it myself). It is not that fast, but when shopping for an all round lens there are typically some trade offs.
You worry about the lens not being full-frame, but since your camera is not a full frame, that is not really a major concern (unless you are thinking about going full-frame later). What matters is the image quality it produces on the camera you have.
It is neither an L-series lens, not because of lack of image quality, but simply because the L-designation is reserved for full-frame lenses. It has the image quality to match the L-series lenses. In fact, AFAIK, this lens was produced as a natural kit lens for the 7D to match the enhanced resolution of the sensor when it was launched, and to meet the high demands of the type of user who would choose the 7D.
If you want to go for a full-frame L-series lens, two options come to mind: 24-70mm f/2.8, and 24-105 f/4 IS. I have the latter and I am extremely happy with it. But it is not as wide which I miss sometimes (the 10-22mm is on my wish list as a supplement). And if you want to go wider on the L-Series lens lineup, you have to go to a dedicated wide angle lens, which is not really what you are looking for.
So all in all, the 15-85 is probably one of the best all-round lens for the 7D. If you don't require that wide a shooting range, and you want the best possible quality, I can recommend the 24-105. If you require a faster lens, and want to go wider than 24mm, then the 17-55mm f/2.8 would be a good bet, as suggested by others. But that has a more limited focal range on the long end.
So to sum up the contenders:
- 15-85mm f/3.5-5.6 IS USM - The wides of the bunch, but not that fast, particularly on the long end.
- 17-55mm f/2.8 IS USM - Fast, but more limited zoom range.
- 24-70mm f/2.8L USM - Fast, but no IS.
- 24-105mm f/4L IS USM - Faster than the 15-85 in most of the zoom range, but slower than the other two, but with IS, and the longest focal length.
So which to choose, depends on your shooting style and requirements.