Caveat: MTW stalled me for the longest time by omitting two points:
- Tensors (as well as exterior and symmetric powers) have very little to do with differential geometry (and a fortiori physics) as such except for motivation—go study multilinear algebra first if the fanciful pictures of egg crates don’t help. Learn the coordinate-free definition of the determinant and rederive Cramer’s rule while you’re at it.
- Nobody can satisfactorily explain what the Riemann tensor is as a whole, though you can learn to work with it and understand some of the direct summands. (“A little monster of (multi)linear algebra”, as Gromov’s refreshingly frank book[1] puts it.) This is one of the rare cases where the worse meaning of von Neumann’s oft-abused quote is in force.
If you like the geometric viewpoint proselytized in the earlier parts of MTW before it gets to GR proper, try also Modern Classical Physics by Thorne and Blandford, which is that in doorstopper form (an early version[2] is available online though not advertised).
Gravitation - Misner, Wheeler, Thorne
Already mentioned: (it's a goody)
Linear Representations of Finite Groups - Serre