- Gravitation by Misner, Thorne, and Wheeler
- This book presents an alternative formulation of tensors as abstract objects. In some ways, this presentation is more natural because it avoids defining tensors in terms of an arbitrary basis, but most physicists seem to use the component notation. For more info see the wikipedia page on Tensor (intrinsic definition)
- Introduction to Tensor Calculus
- This is a great introduction to tensors.
- An Introduction to tensors and group theory for physicists
Another grad student recommended this book. I haven't read it, but there is a free online version for students. From the Amazon page
An Introduction to Tensors and Group Theory for Physicists provides both an intuitive and rigorous approach to tensors and groups and their role in theoretical physics and applied mathematics. A particular aim is to demystify tensors and provide a unified framework for understanding them in the context of classical and quantum physics. Connecting the component formalism prevalent in physics calculations with the abstract but more conceptual formulation found in many mathematical texts, the work will be a welcome addition to the literature on tensors and group theory. Advanced undergraduate and graduate students in physics and applied mathematics will find clarity and insight into the subject in this textbook.