Hello j000e and welcome to the forums.
Now I think you may need to clarify a little further just what you mean by camera speed, as this could be anyone of a number possible things. My guess is, you're talking about burst rates and shutter lag times here, but I could be wrong.
Burst rates are basically how many shots a camera can take per second when set in the sequential shooting mode as opposed to single shot mode. Most DSLRs will manage at least 3 shots per second and some professional models can attain speeds of 7 - 9 shots per second. This is all down to the fact that DSLRs firstly have a much larger sensor than a compact camera, they also use a different type of mechanism to fire the shutter which again reacts much faster than a compact and they also have more focus points to speed up the autofocusing too.
Now with compacts you will also read about shutter lag times, which is basically how long it takes the camera to set the focus, aperture, ISO and white balance to correctly expose the image before it fires the shutter. Some compacts are considerably faster than others. 0.3 of a second is the average time it takes but some compacts will do this in under 0.2 of a second whilst some of the most basic models can take 0.5 of second to accomplish this.
So what are you wanting to photograph that requires either a fast burst rate or shorter shutter lag time to produce a successful image, because this will direct you towards what type of camera you'll need. Now if a compact is going to come up short for this task and you cannot afford to run to a DSLR then you could consider what is termed as a bridge camera. This is like a DSLR, but you can't interchange lenses, for example cameras like the Canon SX1IS or Panasonic FZ38 or Nikon P90. They all have a larger sensor than a compact, but still smaller than a DSLR has and full manual controls. They will also shoot video too, often in full HD as well.