лезть/лазить(лазать) is more like crawl, i.e. often onto horizontal surface. Little babies do it in special clothes:
https://www.google.com/search?q=%D0%...w=1253&bih=779

And yes, it is possible to лезть up and down, and залезать/слезать will do, but -beware - there are heaps of synonyms.