To avoid this movement affecting the image, many cameras allow for the mirror to be locked up out of the way in advance (which also is useful for cleaning the sensor). In this mode, the first depression of the cable release will lock up the mirror; the second will trigger the shutter. A good cable release has a second advantage of allowing the shutter to be locked open in the "B" (bulb) position for exposures longer than the camera's longest timed exposure.
Not all tripods are created equal. A solid tripod is a must, especially in windy conditions. Some tripods have the ability to have a weight hung from the center column to help add stability. A tripod made of carbon fiber will be lighter without giving up quality and is the best type for travel photography. A ballhead attached to the tripod will offer the most flexibility in terms of camera angle. Make sure to turn off image stabilization on a lens that has this option when the camera is mounted on a tripod.