Is there a good quick introduction to knot theory? I am relatively mathematically savvy so any level is appreciated.
As Michael comments, Colin Adams has a well regarded text called "The Knot Book". Adams has also written a comic book about knot theory called "Why Knot?". It's very humorous but is a genuine introduction to the mathematics involved. This comic book comes with a plastic "rope" that can be knotted, unknotted, and twisted into different shapes.
I think "Why Knot?" qualifies well as a "good, quick" introduction to the topic. Well worth tracking down.
Rolfsen's textbook "Knots and links" is quite nice. It assumes a 1st course in algebraic topology, and is pleasant self-learner text. Plenty of nice exercises.
On the higher-end of the knots textbook world, Burde and Zieschang's "Knots" covers quite a lot of ground in much detail. Kawauchi's "A survey of knot theory" covers much more ground but in less detail. Hillman's "Algebraic invariants of links" is more specialized and tends to focus on ideas such as Alexander modules, but it goes into them in more detail than I've seen anywhere outside of Jerry Levine's papers.
Chuck Livingston has a very nice looking book just called "Knot theory". It appears to have a fair bit in common with Rolfsen's book, in that the central theme appears to be the Alexander polynomial. I haven't read it yet (should arrive in a couple days) but it looks promising.
I love the book "On Knots" by Louis Kauffman. It's got a playful style, yet he develops a lot of deep mathematics. I read this in high school, and I got quite a lot out of it, and as my mathematical knowledge progressed, I got more and more out of it.
My intro to knot theory graduate course used "An Introduction to Knot Theory" by Lickorish. The early chapters on Seifert surfaces and polynomials are quite nice.
Two books I would recommend are:
Knots and Surfaces, David Farmer and Theodore Standford (American Mathematical Society; 1996)
Knots and Links, Peter Cromwell (Cambridge U. Press; 2004).
This answer is a little less popular of a suggestion, but it was how I first learned about knots, and I really enjoyed it. This book is primarily focused on Vassiliev Diagrams, and is currently unpublished, but available through the authors webpages. Yay! free stuff!
Here is a link to the CDbook by Chmutov, Duzhin, and Mostovoy. It is the first link on the page.
I personally found the focus on invariants very useful.
My second favorite would be Rolfsen, as Ryan suggested.
V.O. Manturov, Knot theory, freely available online. The first chapter gives an elementaty introduction. To continue, you have to know basic topological concepts, e.g. fundamental group.
I'd certainly recommend all of the titles presented above; I personally worked from the Colin Adams book.
Here is a YouTube channel which I've found is helpful for those studying elementary knot theory.
Although, in my opinion, it is strongly influenced by Rolfsen's Knots and Links, Prasolov and Sossinsky's Knots, Links, Braids and 3-Manifolds: An Introduction to the New Invariants in Low-Dimensional Topology is nice because there are references to recent articles in the appendices of each section. Moreover, it is the one of the very few books which gives an introduction to Vassiliev Invariant.