I agree with the advice above, that if you have access to a library and/or live near a college bookstore then looking though books is the best way to find what works for you. We all learn different ways, so it is great that there is such a variety of books to chose from. IF that is just not possible, used copies of old editions of many books can be found online for cheap, so you may be able to pick up a few books without spending much.
If you have no background, I recommend elementary linear algebra by Anton - used copies of old editions are cheap and quite good. The free online book by Hefferon is also a good choice, although at a slightly higher level than Anton - you may as well start there since it is free, and has a solution manual as well. It is professional quality, only free! I also like the Schaum outline of linear algebra.
Axler's book that has been mentioned alread is excellent, and I have carefully worked through a lot of it, but it is not useful if you have no background in the subject, in my opinion. Even in the preface he states that the book is for a second course.
Finally, after taking an elementary course in linear algebra, I found that I like the way Apostol presents it in Calculus volume II (volume I has some duplicate chapters on linear algebra, but volume II covers it all and shows how it is used in differential equations and multivariable calculus). It is actually in the same spirit of my notes from when I took the elementary course, although the course I took was not as advanced as Apostol. It would have been too much for me to start with, but it is written assuming you know no linear algebra and I know that many folks have learned from it in the past, so it may work for you. It is quite expensive and different than some treatmets, so I would only go this route if you can borrow it before you shell out the serious money.
By the way, I love Strang's Linear algebra and its applications. I have had the third edition for a dozen or more years. When I took linear algebra we used no book, and did more of the abstract aspects than Strang covers, at the expense of learning nothing about matrix square roots, LU and QR decompositions, etc. I felt like Strang taught me how to use the linear algebra I knew in very practical matrix problems - the kind of problems that do come up in my work. But I agree that it would be a terrible book to learn from for the first time, and has a funny mix of topics to be used for a second course in the subject, too.
Good luck,
jason