Alpine mountaineering includes a huge range of activities. Whilst it defies close definition, an 'alpine route' typically involves climbing on a mixture of glaciated terrain, rock and ice and generally starts in a valley and aims for some form of natural summit. Similarly, the commonly accepted alpine areas are the European Alps, the Rockies of North America and the alps on New Zealand's South Island, but increasingly routes in the greater ranges, the Himalaya, Andes and Antarctica, are being tackled in lightweight 'alpine' style rather than the traditional siege-like expeditions of the first half of the 20th century. Having said that, one of my 'best' routes to date did not take place in any of those areas and did not involve glaciers; yet I would argue that it was an alpine route. The definition is really not important, not to me anyway; the point is that it's climbing on a variety of ground, in many areas of the world and using a wide range of mountaineering skills.
My introduction to alpine mountaineering was pretty abnormal. A typical progression for someone in the UK would be to try rock climbing, progress to Winter climbing in Scotland, including steep ice, and then to head down to Chamonix in the French Alps for a Summer season. In contrast, I was talked into hill walking by a friend when I was 24, decided that I liked the scenery and physical exertion and that Scottish hills in Winter would provide more of both and so signed up for a short winter mountaineering course in Scotland. After that I was thoroughly hooked and the following Summer my hill walking friend and I hired a British mountain guide and enjoyed our first alpine season, mostly in the Bernese Oberland region of the Swiss Alps.
It's worth mentioning that, whilst a great many people would say that it's best to gain skills by being a member of a club and 'working up' by gaining skills from club members, I felt that, having started climbing so relatively late, a "fast start" was required. Given that factor, using courses and guides was, for me, a very efficient way to gain skills rapidly and effectively extend the amount of climbing I'll have time to do. For more information on guiding you could visit the web site of the British Mountain Guides.
Since then I have climbed in every Summer season, largely in the European Alps but including an excursion to the Caucasus (which is still technically Europe but very much not part of the Western range!). I've also climbed icefalls, during most Winter seasons, in the European Alps, Scotland and Canada. (See separate page on this site specifically about ice climbing.)
The web is covered with mountaineering sites of various sorts. I've included below some of those that I've used myself and found useful or good enough to return to.
The Alpine Club, the oldest climbing club in the World, is the only UK based mountaineering club catering specifically for those who climb in the Alps and the Greater Ranges of the world. Their home page contains various information, including how to join, forthcoming lectures at the club in London (which prospective members are welcome to attend) and expedition reports.
The Peakware World Mountain Encyclopaedia is a great new site containing a reference to hundreds of mountains around the world. The site is interactive, allowing visitors to comment on peaks and to add information about favourite mountains. If you're looking for information about a particular mountain, or want to run an automated search for routes of a particular type, or in a particular area, then this is a very useful place to go.
Many mountaineers and hillwalkers I know have problems with their knees as a result of pounding them when descending slopes whilst carrying heavy loads of gear. This is particularly so in the UK, where the use of 'ski' poles is only now becoming more popular. I'm not keen on the idea of knee operations and so use two poles all the time, as does Pete Clinch, but unlike me he's written a remarkably comprehensive page on the 'correct' way to use them which you can find at Pete's Poles Page. If you don't already use poles then it's well worth a visit.
The British Mountaineering Council is the UK representative body which exists to "protect the freedoms and promote the interests of climbers, hillwalkers and mountaineers, including ski-mountaineers". Their web site contains information on access, climbing walls, insurance and training, amongst other things.
If you're going to climb in Alaska then there's information available from the on-line version of Alaska's Mountain Magazine. Once you're there, it's fairly likely that you'll need the services of Talkeetna Air Taxi to get to where you're going and even the Alaska Mountaineering School perhaps.
There are more links on my ice climbing page.