It depends on the blur type. Usually:
Motion blur happens if the camera moves during the exposure. When shooting hand held with a 300mm on an APS-C camera, you should shoot at something faster than 1/500s(*). If the lens features image stabilization you can go down to about 1/100 and still have clear shots. You can also use a tripod or a monopod.
Focus blur happens if the focus is not correct:
- The camera focused somewhere else in the picture
- The camera focus is innaccurate (front/back focus). This can be tested by shooting a fixed target (newspaper taped on a wall) with similar settings at a similar distance.
Zooms have a rather shallow depth of field. Having the lens wide open requires a very accurate focus, things can be mitigated by closing a bit (using the lens around f/8, for instance)
Assuming good focus and rock-solid tripod, this can come from a lens that is a bit soft because it is used at its extremes (focal length and aperture). Try to de-zoom a bit and close a bit.
Using faster speeds and smaller apertures may require to increase ISO, resulting in more noise. Photography is often a matter of choosing your lesser evil...
Then, pixel-peeping one's pictures is always disappointing :)
(*) This comes from the "1/f" rule. When hand-held, a full-frame camera without stabilization, you should use a speed faster that the reciprocal of the focal lengh, for instance 1/250s if your lens is a 200mm. On an APS-C cameras you have to take in account the crop factor and multiply the result by 1.5-1.6 (this usually the "next" speed, so 1/400s). But this rule if for overall sharpness, if you pixel-peep you need even faster exposures.