Valais is a Swiss canton that includes such stars as the Matterhorn and even more hidden jems in the valleys usually going from north to south and finishing with a dead end at the Italian border or almost.
The map below presents the distribution of climbing opportunities by regions, from left to right.
Who hasn't seen Cervin (Matterhorn) on at least one advertisement for chocolate, watches or bras? This mountaineering symbol is on the Swiss-Italian border, accessible from the Swiss side starting at the dead-end village of Zermatt (most commonly to do the Hornli Arrete, the first-ascent route and still the most common and easy way to get to the top). You can find more detail on this and other mountains in the Zermatt and Zinal valleys region.
Further mountaineering exploration can occupy many of your days and nights at the Monte-Rosa mountain chain, also accessible from Zermatt on the Swiss side. As there are so many summits and interesting climbs at this chain, a special region, covered by Monte-Rosa Chain has been devoted to it.
More exploration will certainly bring you to the Verbier village, a skier's and mountaineers' paradise or to Saas-Fe, one valley to the east after Zermatt with much less people and more glaciers and summits. The French border around Martigny and the west corner around Simplon should not be forgotten either as they abound in rock, snow and ice for all tastes.