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I'm studying Riemannian Geometry, and I'm having a lot of trouble with the books Riemannian Geometry and Differential Forms, both from do Carmo.

What book(s) would you suggest? I would also like the book to have examples and calculations, and (if possible) applications to physics. Thank you!

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I think this is a good question, because I was once in almost the exact same situation.

One good introductory reference, in my opinion, is Tensors, Differential Forms, and Variational Principles by Lovelock and Rund, because it gives some intuition on differential forms and moving frames, as well as on tensor theory and differential (including some Riemannian) geometry. It even has some applications to physics. Both of those do Carmo works are a little skimpy on details (again, my opinion), so I found it quite helpful to be introduced to the subjects lightly first.

Afterwards one can move to more focused, but still introductory works: e.g. see these references on moving frames and these references on beginning Riemmanian geometry.

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