How can a senior undergraduate find academic journals in math and computer science?

I would agree that you should mostly be ignoring journals and focusing on conferences, as they publish the majority of new computer science research.

Microsoft Academic Search is a good place to go to get an approximate listing of the top conferences for each sub-field of computer science. Other fields other than Computer Science are listed there too.

Use your school's network to access the ACM Digital Library (this should be free through your school's library), and download the proceedings for the conferences in the past year or two. Find the papers that look interesting to you and then search for them on Google Scholar. You can print them out if you prefer hard copies.

Hope this helps!

@willwest has answered regarding CS. I will answer regarding math.

I would start with the journals of the AMS (pure math) and of SIAM (applied math). These are the pre-eminent professional societies in their fields and virtually all of their journals are top tier. In particular, you might start by browsing the Journal of the AMS and the SIAM Review, the most selective journals from each society.

The journal that a paper gets published in is becoming less and less important, since most researchers find articles through search engines or social media rather than by browsing journals. The best way to keep up with new research in a particular subfield of math or CS is to subscribe to the appropriate arXiv RSS feed; for instance, for numerical analysis this is http://arxiv.org/rss/math.NA. This is how I usually learn about relevant new research.

Note that few mathematical conferences have proceedings, and none that I know of are considered prestigious (in CS, the situation is roughly the opposite). If you want to know which journals are the most highly regarded, talk to faculty in the field.

Journal articles are PDFs, so you can view them with any mobile app that understands PDFs. If you want to read a hard copy, either print the paper or go to your campus library.

Let me put in a vote for Mathematical Reviews (online version). They're a few months behind the actual publication of the article, but someone who knows the field is giving you a two-paragraph (plus-or-minus) synopsis. Even low-quality papers get reviewed, and the authors-should-have-read-X comments of the reviewer are well worth it.

Your department can arrange a login for you. For that matter, just go join the American Mathematical Society; dues for grad students are trivial.