The American Mathematics Society (AMS) website lists many different opportunities for undergraduates interested in mathematics. This is a good place to begin finding out about career opportunities, study abroad possibilities, summer programs and internships, graduate study and so on.
There are many excellent summer research programs. The National Science Foundation (NSF) supports several REUs (Research Experiences for Undergraduates) with different emphases; some on math biology, some just for women, some for sophomores, some for more advanced students... You are paid for participating (enough to cover expenses).
For information visit: http://www.ams.org/employment/reu.html
There are also summer research programs run under different auspices. For example the Columbia Mathematics department has one: for information about summer 2013 visit http://www.math.columbia.edu/programs/main/one/summerprogram.html
The Institute for Advanced Study and the Park City Mathematics Institute jointly sponsor a yearly summer meeting in Park City Utah with an undergraduate component. http://pcmi.ias.edu
The NSF sponsored institutes have a few programs for undergraduates, though their main focus is at the graduate level and above.
You can find them by searching their joint web site: http://www.mathinstitutes.org
There is also a lot of interesting information about mathematics on the MSRI website: www.msri.org
A current Barnard student gives an enthusiastic endorsement of the Carleton College summer Math program. http://www.math.carleton.edu/smp/
Also, it is worth checking out the wide ranging Nebraska summer program in applied math and science. http://www.unl.edu/summerprogram
PROGRAMS FOR WOMEN
There are several well established and very worthwhile programs for women. Again, your expenses are paid (from grants from NSF, NSA and so on). Here are two of the best.
1. The University of Nebraska in Lincoln has been running a weekend conference in lat January for female undergraduate Math majors. There are about 150 participants from all over North America; it is an enlightening and enjoyable experience.
Visit: http://www.math.unl.edu/~ncuwm/15thAnnual/ for information on the Jan 2013 meeting.
2. The Institute for Advanced Study and Princeton University jointly run a 10 day program (next in May 13-24, 2013) for undergraduates, graduate students and postdocs. Subject: combinatorics and graph theory. Visit: http://www.math.ias.edu/wam/2013