I am really confused about the use of tensors in special relativity. I'm not sure if I understood what is a covariant and a contravariant tensor and vector. I am looking for a book of relativity that teach this.
Schutz's A First Course in General Relativity does a good job about tensors and relativity in the first few chapters.
He has another book dedicated to specifically the mathematics, but its title slips my mind. If its as good as those briefer chapters, you won't go wrong.
A more thorough treatment of the mathematics can be found in Tensors, Differential Forms, and Variational Principles
Almost all the standard books on general relativity include a section on tensors and special relativity.
A First Course in General Relativity, 2nd ed., B. Schutz, Cambridge University Press 2009. This is an excellent book containing tensors and STR.
An Introduction to General Theory of Relativity by J. V. Narlikar provides a general introduction on tensors and special relativity.
General Relativity: An Introduction for Physicists : A textbook by Anthony N. Lasenby, George Efstathiou, and M. P. Hobson includes a chapter on tensor calculus on manifolds along with certain exercises.
Tensor analysis in special relativity is a freely available PDF with good exercise problems.
Spacetime and Geometry is a graduate-level textbook on general relativity by Sean Caroll. It offers a chapter on Special Relativity and Flat Spacetime. Includes a necessary description of tensor algebra.
Only for tensor calculus:
Schaum's Outline of Tensor Calculus: A textbook by David Kay covers all fundamental concepts along with 300+ fully worked problems and solutions. This can be supplemented with standard textbooks.
Introduction to Tensor Calculus: Freely available on arXiv. This can be used as a short reference for an introductory course on tensor algebra and calculus.