Lenses for dSLRs are much more complicated than lenses for P&S digicams. The lens has to be (relatively) far away from the image sensor in order to make room for the mirror swinging out of the way. Also, because the sensor is physically larger, so must the optical elements. So lenses that cover the same (relative) range of focal lengths as the lenses on the super zoom cameras you are accustomed to, are quite large, quite expensive, and (unfortunately) quite flawed. The Tamron AF18-250mm F/3.5-6.3 Di-II LD Aspherical (IF) Macro, arguably the best lens of its kind, is soft throughout its zoom range, and suffers from optical distortion at its wide end and chromatic aberration at its long end. Add to that it is almost a full f-stop dimmer that what you're accustomed to with your SP-570.
But remember that one of the benefits of dSLRs is that they can be customized to be the best possible photographic instrument for a particular purpose. So an "all-around lens" would, to a significant extent, obviate many of the the advantages dSLRs have over P&S digicams. Two or more lenses with less ambitious zoom ranges, would provide better image quality and be less expensive.
Canon makes many fine dSLRs, and enjoys a wealth of OEM and third party lenses and accessories. But remember that, with a dSLR, you don't just buy a camera, you buy into a system, and you need to pick the system that will serve you best. One component of this is how you want your image stabilized. Super Zoom P&S digicams are either stabilized or they're not. When selecting a dLSR, you must pick the flavor of stabilization you want. Canon and Nikon use optical image stabilization in some of their lenses, which makes them bigger, heavier, and more expensive. Pentax, Sony and most Olympus dSLRs use sensor shift image stabilization in the camera body, so any lens attached to the camera will be stabilized. That includes used lenses manufactured over 20 years ago. The market for stabilized Canon or Nikon lenses is new, so there aren't very many on the used market so far.
With P&S digicams, you can buy a camera from one manufacturer, and when you want to upgrade a few years later, you can buy one from another manufacturer. That won't happen with a dSLR. When you buy a dSLR, the camera body might actually end up the smallest part of your investment. When you want to upgrade your camera, it must be from the same manufacturer or your collection of accessories and especially lenses won't work on your new camera. You need to pick your system carefully, not just your camera.