Having selected the country you plan to visit, the next requirement is for some research. Multiple resources are available to help with this. This page provides some useful links, but Alpine Club members are also a ready source of expedition advice and information.
Searching the premiere journals - the American Alpine Journal, the Himalayan Journal and the Japanese Alpine News - for the region of interest is a good place to start. The first two of these are available online from their respective clubs: the American Alpine Club; and the Himalayan Club; and both are readily searchable.
The Alpine Club also provides a number of resources that will be useful for those planning or researching expeditions:
The Alpine Journal is a substantial annual record of mountaineering achievement, if you are planning an expedition it should be on your reading list.
The Mount Everest Foundation maintains a large online archive of all of their funded expeditions, from 1954 to the present day.
Alpine Club Expeditions receive funding through the Club and an archive of their reports are held on the Club website.
The Royal Geographical Society holds copies of all MEF reports as well as many others. Searching their online archive is very straightforward and summaries are provided. Reports cover the period 1965 onwards and are very comprehensive. Copies can be consulted by visiting the RGS library, or digital versions can be e-mailed.
The National Archives at Kew hold many older documents deposited there from multiple sources.
The British Mountaineering Council is another source of funding for expeditions. Many expeditions supported by the BMC will also have been supported by the AC and/or the MEF, but there are occasional reports in their archive that do not appear elsewhere.
The Al Rouse Collection at Sheffield Libraries holds a comprehensive archive of mountaineering material, including copies of MEF and other expedition reports. The collection can be searched on the library catalogue by selecting 'Rouse Climbing Collection' from the filter menu.
In order to search for specific mountains, the Himalayan Index, which is maintained by the Alpine Club, is another important resource. If you are lucky, the Alpine Club photo library may even have a picture of your intended objective and it is always worth contacting them to check.
For more popular areas, such as the Andes, excellent guidebooks are available that provide not only route descriptions, but also useful local information.
With respect to the Indian Himalaya, Harish Kapadia, a recognised authority, has published a large number of books that provide invaluable information on the many areas that he has visited.
For China and Tibet Tom Nakamura has produced a variety of articles and publications detailing unclimbed mountains in the Nyainquentanglha range and Sechuan.
Once you've made full use of all of these resources, making contact with the leaders of past expeditions to the area that you have selected is the next step in choosing objectives appropriate to your requirements.