Unfortunately I don't know any quantum stuff. I learned them for machine learning purposes.
A monograph by Cichocki et al. (part 1, part 2) is an overview of how tensor decompositions, tensor formats, and tensor networks can be used in machine learning and signal processing. I think it lacks some applications, including acceleration and compression of neural networks by compression of weights of layers using tensor decompositions (this also sometimes improves accuracy, probably by reducing overfit).
Tensor decompositions and Applications by Kolda, Bader 2009 - this is an overview of tensor decompositions. It doesn't have many machine learning applications. Also it doesn't talk of tensor networks, only about some simplest tensor decompositions and specific tensor formats which are the most popular types of tensor networks. This paper was the first thing I read about all the tensor stuff, and it's one of the easier things to read. I recommend you read it first and then look at the topics that seem interesting to you in Cichocki et al.
Tensor spaces and numerical tensor calculus - Hackbusch 2012 - this textbook covers mathematics of tensor formats and tensor decompositions for hilbert and banach spaces. No applications, a lot of math, functions analysis is kinda a prerequisite. Very dense and difficult to read textbook. Also doesn't talk of tensor networks, only about specific tensor formats.
Handwaving and interpretive dance - This is simple, it's about tensor networks, not other tensor stuff. It's for physicists but chapter 1 and maybe other chapters can be read without physics background.
Regarding the TensorNetwork library. I've skim-read it. I haven't tried using it. I think it's in early alpha or something. How usable it is for me depends on how well it can interact with pytorch and how easy it is to do autodifferentiation w.r.t. core tensors and use the tensor network in a pytorch model. Intuitively the API seemed nice. I think their idea is to that you take a tensor, make it into a matrix, do truncated svd, now you have 2 matrices, turn them back to tensors. Now you do the same for them. This way you can perform some but not all popular tensor decomposition algorithms.
P.S. Fel free to message me if you have questions about tensor decomposition/network/formats stuff
In light of reading Hazard's Shortform Feed -- which I really enjoy -- based on Raemon's Shortform feed, I'm making my own. There be thoughts here. Hopefully, this will also get me posting more.