So first let's explain what ski touring is. It's not the same as cross-country skiing, and not quite the same as telemarking. It's a way of skiing both downhill and uphill, the key being that the skis have removable "skins" (nowadays synthetic) which you can stick on for the uphill sections, and special bindings which can lift the heel up for uphill bits or can lock it down for downhill bits.
It's a very different experience from normal downhill skiing in a ski resort, you get to experience the wild wintriness away from the crowds, at a much slower pace (more like hiking), but without the queues and machinery. The climb is obviously most of the time spent, and the off-piste descent in fresh snow is the reward for the effort put in.
A tour can be a simple climb and descent on the same route, or can make a day loop. You can even use the huts which are open in winter to make a tour of several days, but day tours, usually starting early morning and finishing in early afternoon, are more common.
The equipment is slightly specialised, the skis look very much like normal downhill skis except they're not quite as strongly shaped. The bindings as mentioned are special in that they are hinged at the toe and the heel can lock down or move free. The back of the binding can be set so that even on steep uphill climbs, the footstop keeps the foot horizontal. And the boots are more flexible than normal ski boots, and don't lean forward as much. In addition you need the removable skins, and if there is the possibility of ice then you can also attach metal spikes to the binding like crampons. The ski sticks are just normal sticks, they can either be fixed or telescopic.
Ski tours don't have to be done with skis, it's also possible to do the same tours with snowshoes and snowboards. This has the advantage of being able to ride the deep off-piste snow with a floaty snowboard rather than sinking skis (especially advantageous if you're more comfortable on a board), but does have disadvantages. Firstly, you have to carry the board on the uphill bits and carry the snowshoes on the downhill bits - this makes for extra weight and extra effort. Secondly, switching between snowshoes and board is more awkward than just applying and removing the skins on the skis. It depends on the tour, if it has alternating uphill and downhill bits, or long flat sections then it won't be as suitable for the snowboarders.
As far as clothing is concerned, normal skiing or snowboarding gear will do, but layers are useful to adapt to the temperature. A helmet is advisable for the descent but uncomfortable for the ascent - so you may find yourself needing hat and sunglasses for uphill and helmet and goggles for downhill.
Because the ski tours go off the beaten track in the snow-covered mountains, there is always the risk of avalanches to consider. Therefore it's not recommended to go alone, and additional avalanche equipment such as transponders / sensors for finding and being found, and shovels for the digging, are highly recommended. Of course the equipment should be tested and you should know how to use it all.
By far the best way to start out is to find a local club and ask them when they have a beginner-friendly tour. You may be able to borrow or rent some of the equipment from the club itself, and the rest you can rent from a shop. For example, in Zürich there are a couple of outdoors shops which let you hire equipment for the weekend, including touring skis, boots and sticks. Ruedi Bergsport is one recommended example.
If the club welcomes beginners, they'll be able to explain the equipment and how to use it, and will guide you on a tour of a suitable level. They'll also have the experience of mountain safety and assessing the prevailing avalanche risks. Membership of such a club brings other benefits too, as well as a friendly social atmosphere.
Important is to know what the first tour is like - the metres climb (in our case, 700m up and 700m down) is comparable with day hikes, to get an idea of what fitness is required. Obviously the snow requires some extra effort but the equipment is light and the pace can be set accordingly.
Starting the ascent
Zig-zagging up the slope
Nearing the summit
Off-piste reward for the climb
The basic techniques aren't difficult to grasp, the climbing is straightforward as long as you try to slide the skis forward rather than waste effort lifting them up and putting them down again. The heel stops help when climbing a steeper gradient, but zigzags can keep the steepness to a steady comfortable level. The only tricky bit about zigzags is making the switchbacks on steep terrain - this requires a little bit of gymnastics to swing the uphill ski round (to become the downhill ski for the next bit) and then somehow feed the other ski round too. With the hinged bindings the tips can be awkward to lift up but this just takes a bit of practice.
When it starts getting tricky is on steep traverses, when the ski edges may struggle to grip the slope, and on icy sections. Fortunately our snow conditions were near-perfect so we never got to try the "Harscheisen" (metal spikes used for extra grip). Again, local knowledge of the mountains and snow conditions are essential to choosing a suitable route.
For the descent, the skins are removed and the bindings locked down, the boots switched to locked mode and then they really behave just like normal downhill skis. If you're lucky you'll get untracked soft to explore, far from the tracked-out resort runs, but obviously this is strongly dependent on the season - many ski tours are run in spring when the snow is far from fresh or soft.
For a good introduction, see Wikipedia's page on ski touring.