Resources

Undergraduate Research

List of Mathematics REU's

List of Mathematics REU's Compiled by the NSF

The NSF also has a list of REU's in other scientific fields. You can select a field here.

Also bear in mind that your professors tend to know a thing or two about research, asking them for suggestions is okay.

Online Resources

Paul's Online Notes

A well-written set of notes covering calculus, the basics of differential equations, and high school algebra. About the same quality as any of the textbooks on this subject, and they're free.

Purdue Problem of the Week

Has a huge list of difficult problems.

Proof Wiki

Has a surprisingly large collection of proofs. Much easier to look here than hunt through the library for a specific proof.

Math Stack Exchange

Online community for answering basic math questions. If your question pertains to undergraduate mathematics, you should ask it here.

Math Overflow

Similar to math stack exchange, but for advanced mathematics, e.g. research level mathematics.

Books that Are Actually Useful

General Topology

A reasonably accessible introduction to general topology. Covers all of the basics. Has quite a lot of content, and it's quite cheap.

Algebraic Topology

Relatively easy to read.

Advanced Linear Algebra

A genuinely enjoyable read. Covers the theoretical aspects of linear algebra.

Measure Theory

One of my favorite books. Measure theory is very interesting, and Halmos is an excellent writer.

Advanced Calculus

Try chapters three and four for a good minicourse in analysis. The rest of the book focuses on vector space calculus and manifolds.

Real and Complex Analysis

Rudin is pretty well known as a writer. The book's good. I suggest reading a book on measure theory first, because he moves very quickly at the start. You can find cheaper versions than the one here. Bear in mind that this book covers exclusively graduate analysis. If you want an introduction to analysis, look elsewhere.

Books for Fun

A Course in Arithmetic - Jean-Pierre Serre

An easy introduction to arithmetic. Worth reading even if you already know arithmetic.

Basic Number Theory - Andre Weil

Another easy introduction to one of the core fields of modern mathematics.

Basic Algebra I and Basic Algebra II - Nathan Jacobson

The last of the three "Textbooks with Incredibly Misleading Names." Still pretty good, as mathematics books go.

Inter-Universal Teichmuller Theory I, II, III, IV - Shinichi Mochuzuki

Good luck.