This is a fairly common problem to run into. My guess is that your SRTM data are still in geographical coordinates and has not been reprojected. As such, the elevation units are being measured in metres while the xy-units are in degrees. You need to use the z-conversion factor to convert the xy-units and z-units so that they are the same. The common approach is to use this formula to calculate an appropriate z-conversion factor:
zFactor = 1.0 / (113200 * cos(midLat))
Where midLat is the middle latitude of your study area. Consider as an example the following SRTM DEM, still in the original geographical coordinates. The first image uses a z-conversion factor of 1 (the default) and the second uses a modified z-conversion factor based on the equation above:
Then, adjusting the display parameters (e.g. min/max values that the colour ramp is scaled to) can be used for further fine-tuning until you get it just they way you want:
The reason that the z units and the xy units must be the same is that hillshade algorithms need to calculate the slope gradient. "Rise over run" (i.e. tan slope) isn't that useful as a measure of slope gradient when the rise and the run are measured in different units. An alternative solution therefore would simply be to project the data into a reference system that uses metres as the xy-units.