Walking into a hiking shop these days can be a daunting experience. So many different products on offer, so many branded fabrics with special coatings and technical whatnots, and if you're not careful you can end up spending a fortune. But it needn't be so. Decide what kind of walking you're planning to do, and that'll help determine what bits of kit and what you don't (yet). If you're just going on Sunday afternoon strolls, you don't need the full works!
For example, to do some of the walks in Switzerland, ordinary trainers are often fine for the easy or very easy ones. However as soon as you get rough tracks, steep paths or slippery conditions, it's best to get a pair of boots designed for the purpose, with sturdy, grippy soles, ankle support, and a reasonably tough construction. But even here you don't need to spend a fortune, a budget brand will serve equally well provided it fits properly and is comfortable. Try them on in the shop, making sure you wear the same thickness of socks (preferably the same socks) that you'll wear when walking. If the shop has a small 'rocky ramp' to try them on, then make use of it - see how the boots feel going uphill or downhill.
Obviously, the budget brands don't meet everyone's needs, and that's when the famous names come in with their higher quality manufacturing, exotic fabrics and coatings, and other features. Magazine reviews will help to sort the corkers from the waffle, and tell you what to look for.
Make sure your jacket is waterproof, and make sure your trousers aren't denim. Additional layers can provide windproofing, and waterproof overtrousers can protect you if the weather surprises you, but the most important factor is comfort.
What you take depends on how long you're going for, but a rucksack of some description is the best way to carry what you need. At the very minimum, it needs to hold spare clothing, drinks, sun protection and camera, and there are hosts of day packs on sale with a suitable capacity. Important factors are the comfort of the shoulder straps, the ease of access, and the size. Beyond that, look at waist straps to even out the load, and back ventilation systems if they're important to you. Note that some of these systems make the pack quite heavy and prevent it from being folded up inside other bags.
Some packs have special compartments for drinks containers, with the drinking tube passed through to the shoulder straps. This is a matter of taste, but can be extremely convenient.
Some people find them extremely helpful, some people find that they just get in the way. Their popularity has grown enormously in recent years, and there are now many types to choose from in many shops. The idea is that these light, telescopic poles, which look similar to skiing sticks, allow you to transfer weight and shocks to all four limbs, rather than just your poor knees, and hence make walking easier and more efficient. They also provide additional stability and confidence on trickier sections, especially rocky downhill bits, so some people find them invaluable.
As to the question of one or two, that's another open question. Try to borrow some from a friend and try them out before you buy - and remember they may take some getting used to.
Crampons sound like something intimidating that mountaineers use when they're dicing with death, but these mini-crampons are far less dramatic. They come in a few different varieties, but are essentially just small spiked attachments which clip on to the heels or soles of your boots. They're cheap, light, fairly easy to use (as long as they're very securely attached), and for Winter walking, they can be brought out if the path gets icy. The small metal spikes just give more grip on the slippery ice, which in some cases can be invaluable. Hopefully the icy section is limited, and then the spikes can be taken off again. Available from walking / skiing equipment shops, but probably only in Winter.
Another winter-only accessory, snow shoes also come in a variety of styles, most having a large plastic base and a hinged plate to which your boot is strapped. The wide base stops you sinking in the soft snow, and the metal spikes on the base and on the toe give plenty of traction, allowing you to make short work of what would otherwise be difficult walking. They're surprisingly expensive though, so consider rental before taking the plunge. If you're exploring exposed places with snowshoes, it's also a good idea to be aware of the avalanche risk and go suitably equipped (for example, with avalanche transceivers and shovels). Courses are available to teach you how to use this stuff and how to avoid having to use it.
A GPS receiver looks a bit like a mobile phone with a larger screen, and by picking up signals from orbiting satellites, it can calculate where you are. It's no substitute for a map, but it can provide very accurate information to use alongside the map. It can also keep a record of where you've been, including how high, how far, how fast and so on, which you can play about with afterwards. See the Swiss hikes for some examples of the 3d pictures you can make.
You can also take the tracks that you've previously made (or other people have made), and load them in so you can follow them, find a remote campsite given the coordinates, find your way back to your car, use it in treasure hunts, or whatever. You can even use it as a speedometer when you're snowboarding! [Play safe, kids]
A common misconception is that using a GPS makes your position public, in a "Big Brother" kind of way, but that's not the case - the satellites transmit the signals, your receiver receives them, but there's no way for the satellite to tell where you are. Those of a paranoid disposition should be far more concerned about their mobile phones, which allow the base stations to track movements very precisely whether they're making calls or not.