Jeevanjee - An Introduction to Tensors and Group Theory for Physicists
Modern and example-motivated introduction to tensors, but lacking the traditional definition-lemma-theorem structure
Bowen and Wang - Introduction to Vectors and Tensors {???} Traditional mathematical textbook that covers linear and multilinear algebra and then tensor analysis ab initio.
XylyXylyX - What is a Tensor?
Easy introduction that assumes very little previous knowledge and proceeds with intermediate rigour to cover a large part of tensor analysis