Most of the output from mathematical research ends up in a mathematical journal. However, there is a multitude of such journals, and it can be hard to know which ones are most useful. In between my attempts to write a PhD thesis, I have been trying to get some idea of how maths journals work, which ones are good, which ones you can access online etc, and this process has now resulted in a resource page on maths journals. First of all, there is a list of around 45 journals which are my personal favourites, and I think this should give a good idea of which journals are most interesting, if you share my mathematical interests (arithmetic and algebraic geometry, algebraic topology, K-theory, category theory, and in particular cohomological and homotopical methods in these fields). I have included nonspecialized journals such as Proceedings of the AMS and Duke Mathematical Journal, whenever I feel that they tend to publish a significant number of articles I find exciting, and I believe I have included all high-quality journals specializing in the subfields just mentioned. If your interests are only partially overlapping with mine, or if you want to explore journal beyond the ones I have listed, it should be fairly easy to make your own list of favourite journals. Just throw out the irrelevant specialized journals from my list, and have a quick browse through the journal sections of the major journal publishers/providers – there are direct links to more or less all of them at the bottom of the resource page. Most journals have fairly informative names, although the Topoi journal from Springer did not quite live up to what the name seemed to promise.
All in all, there seems to be somewhere between 500 and 3000 mathematical journals, depending on what you count as a journal, and depending on what you mean by mathematical. Some of these are freely accessible to everyone, and for some you need to either subscribe or pay for each article individually. Many journals have a moving wall, meaning that you need to pay in order to access the most recent volumes (typically the last 5 years), while earlier volumes are free. If you want to know when a new issue is published, most journals have email alert services for this purpose.
By now most journals are completely digitized and available online, provided you (or rather your university library) have the right subscription. Of the journals in my list, I think the only exceptions are Journal of Algebraic Geometry, Comptes Rendus, and Memoirs of the AMS. The Memoirs will become available from early 2010 according to the AMS web page, although it is not clear whether this will include only new volumes or all previous volumes as well. The Journal of Algebraic Geometry volumes are available online starting from 2002, but the AMS Customer Service is not aware of any plans to digitize older volumes. As for Comptes Rendus, ScienceDirect provides access back to 1997, and hopefully earlier volumes will become available eventually. When it comes to book series which to some extent resemble journals, the Lecture Notes in Mathematics recently became available through SpringerLink, which is terrific, provided your university has a subscription. The early issues of the Bourbaki seminar are supposed to come online for free during 2010 at NUMDAM, but for the many wonderful Astérisque volumes, there are only tables of contents online, and as far as I know no plans for digitizing in the near future.
Many journals are available from more than one source, and sometimes one source is free while the other is restricted-access and very expensive. A notable example is Inventiones volumes from before 1996, which are available from Springer through subscription or by paying $34 per article, but are also available for free at the DigiZeitschriften site. The AMS Digital Mathematics Registry has a quite exhaustive list of journals where you can see all online sources for any given journal, which is sometimes very convenient.
There is a lot one could say about journal pricing, the serials crisis, the open access movement, and the evil deeds of certain publishers such as selling complete nonsense or hidden advertising under the cloak of a scientific journal. However, I will only refer to the EUREKA journal watch site, which has lots of material and many excellent links. It is interesting to realize that even a big and relatively rich university such as Cambridge can only afford to provide very partial access to the online mathematical literature. For example, I cannot access major journals such as Proceedings of the London Mathematical Society, Duke Mathematical Journal, or Journal für die Reine und Angewandte Mathematik, unless I get access through some other university.
There are various so called metrics which are supposed to measure the quality of a journal, or the quality of an individual researcher’s output. Other people have already written about why the Science Citation Index and the Impact Factor are stupid and dangerous. There exist attempts at constructing better measures, for example the eigenfactor metric.
To find specific articles you are looking for, the most efficient tool appears to be Google Scholar. Although still in beta, it almost always gives you what you want if you search for the author name and some keywords from the article title, for example the phrase “absolute Hodge author:Beilinson“. It has also become quite good at finding direct links to freely available preprint versions of journal articles with restricted access, and it has links to Google Books, which means that you can sometimes get to read articles from various conference proceedings, which otherwise typically exist only in paper form.
To organise articles you find online, there are various pieces of software, such as Endnote, Zotero, and Mendeley. I have only tried Mendeley, and although the idea looks absolutely brilliant, it did not work very well. In particular, almost all the automatically retrieved metadata was completely wrong, expecially for articles in French and older articles. If anyone has positive experiences with Zotero or some other similar software I would be very interested.
Finally, I would like to recommend this incredibly well-written blog post by Michael Nielsen, discussing the future of scientific publishing.
There are also other links at the journal resource page. I wish you a good summer and happy journal browsing!