In reply to David Coley:
Tom's suggestion around Saas Grund is great. I vaguely remember the Portengrat (sp??) being a fun rock mountaineering route too from the Almagellar Hut. There are also plenty of accessible ( arguably too accessible ) peaks around Saas Fee.
I have used Zermatt before. There is cheap (ish ) camping in Randa - a village before the road restrictions. They offer a cheap minibus to Zermatt. You can actually walk on trails to 3,200 or 3,400 metres plus such as the Rotthorn, Fluehorn ridge and beyond. Awesome views and a great first hike. The trail up to the Hornli Hut at 3,200 metres may inspire some ( I don't recommend staying the night though ). Perhaps could even play on the first section of the Hornli route before descending. Although a bit of a jump in height taking the cable car to Klein Matterhorn gives very easy access to a glacier, very easy and short 4,000 metre peak ( Breithorn ) and lots to explore around there. Depending on how people feel the Ayas hut at 3,400 metres gives lots of options, even if it is just glacier practice if required. Strongly recommend the half traverse of the Breithorn and then options for Castor / Pollux.
I'm sure others will come along with great route suggestions...