To dream of climbing is to dream of getting away from something, possibly to escape. We may be avoiding trouble.
Climb Ascension, in the sense of climbing to achieve enlightenment, is an often perceived spiritual symbol.
Climb Climbing upwards suggests we are trying to reach new heights in our lives, and are possibly having to make greater efforts than before to succeed. Climbing down, in the sense that it uses a different technique, suggests having to assess the risks of a change in position.