An easy to read (minimal theorems) but authoritative and modern introduction to tensors can be found in the book “An Introduction to Tensors and Group Theory fo

An easy to read (minimal theorems) but authoritative and modern introduction to tensors can be found in the book “An Introduction to Tensors and Group Theory for Physicists” by Jevanjee (https://www.springer.com/gp/book/9783319147932 https://www.springer.com/gp/book/9783319147932). He takes the more geometrical perspective as a tensor as a multi-linear function of vectors, from which all other statements about tensors (eg how the components transform) follows straightforwardly. Lots of other great material in this book and best of all there are loads of examples.