In other words, how to track down very particular pieces of infomation or answers...Obviously one can't read the entire lot every time one has a query! Is there some online resource that makes it easier?
Your best source for Papal Encyclicals (official statements of the Pope ex cathedra)
comes from the Vatican itself comes from EWTN. Check out: http://www.papalencyclicals.net/. The official http://www.vatican.va/phome_en.htm houses several documents, but the search interface is more difficult.
Additionally, while not an official source, the Catholic Encyclopedia tends to be a good summary of issues from an informed theological perspective.
One of the nice things about Catholicism, from a scholarly perspective, is that it is incredibly well documented.
Knowing what to look for
A great online starting point is the Catechism of the Catholic Church, available from the Holy See and from the US bishops. It is thematically organized and has copious footnotes which point you to more thorough doctrinal sources (the abbreviations need a bit of interpretation - check the appendix).
Looking up original sources
Affable Geek has pointed out the official Vatican site, as well as an archive of encyclicals. Here are a few other resources I know about.
The reference work Sources of Catholic Dogma is available online in its 1955 edition; it contains translations of various original sources, organized chronologically. Obviously this does not cover anything after 1955, such as the Second Vatican Council. It is in English and is full-text searchable.
The website Documenta Catholica Omnia is very useful, if
- You know what you are looking for (author/title),
- You want to read it in the original language, and
- You can read Latin, which is the language of the site interface.
It is not particularly searchable, since many of the documents are immune to OCR. But it is very comprehensive - it doesn't have everything, but it does have an awful lot.
Christian Classics Ethereal Library has online versions of many important works in translation, and it is searchable. It has (I think) the whole of Philip Schaff's Ante-Nicene Fathers and Nicene and Post-Nicene Fathers, in English.
Google Books is actually quite good as well for finding all sorts of treasures in full-text search. If you find a snippet quoted somewhere, and want to find the full-text source, just plop it into Google Books and see what pops out.
Also, I don't know where you are in the world so I can't give specific guidance - but there are plenty of things in libraries that aren't online. Sometimes we just have to go there.