If anything in this article becomes too technical then read the Photography 101 article first.
Mountain photography involves photography of both near and far and all season long, but outside of late spring to early summer is when these giants are most interesting because of the storms and snowcaps.
Mountain photography is NOT wildlife photography. A 500mm lens is very rarely useful. The most useful lens range is 24-200mm and 300mm is not out of the question on a full frame camera. For an APS-C sensor-sized camera then an 18-200mm lens is adequate. The wide-angle will allow for capturing images while amongst the mountains and the longer focal lengths will allow for capturing mountains from afar. If not amongst the mountains then a 70-200mm zoom plus a 300mm lens or just a 70-300mm zoom is a good place to start.
Do not worry about bringing a UV filter because all modern digital cameras have one built-in to the sensor, but a polarizer is a good idea. Remember to rotate it to the desired saturation when capturing the image.
Use a modern digital camera and expose for the highlights (sky) and in post-processing, increase the exposure, lower the highlights and then increase the shadows. The degree to which these adjustments will need to be done will vary with every photograph but this is the general procedure. The alternative is to bring a neutral density graduated filter, but these don't generally fit mountain scenes too well with all the peaks and valleys.
When everything is snow covered, some camera light meters can be fooled and will underexpose the image. Make sure to check the image's histogram for underexposed shadows.
Try to include something in the foreground to add interest (make sure to use a small aperture for greater depth of field) and it helps to include something for scale be it man-made or a tree, etc. With this the size of the mountain becomes obvious to the photo's admirer. And importantly, try to capture the mountain(s) during sunrise or sunset for an added pop of color.
Don't be afraid to try panoramic shots of mountains as this is a great way to isolate the mountain range and not to emphasize the sky and foreground. Here, three images were stitched together using Adobe Lightroom.