Thanks for the posts and references. I want to issue a warning though on some of these. the free notes available at
http://www.math.odu.edu/~jhh/counter2.html
for example, seem devoted primarily to the old fashioned, index - dominated non conceptual approach, ignoring the meanings of the objects being discussed, at least in the first chapter I read, introducing tensors. maybe it gets more conceptual later.
This "indices only" approach however has been out of date for over 40 years in mathematical circles, and it was my impression also, that even in the book on relativity by the physicists J.A. Wheeler and Kip Thorne, written in the 70's, the modern viewpoint was used.
The site at wikipedia suggested by someone in the thread What is a tensor?, has both approaches, and tries to bridge the gap, while pointing out that the conceptual approach is pretty much the standard in advanced work nowadays.
Bear in mind that I am a mathematician, and not a physicist, but you might check out which language the modern physicists are using more today and try to learn that.
One remark is that the conceptual does allow you to understand the indexed approach eventually, since it is just the coordinate representation of the conceptual version, but the index only approach seems to make it very hard to understand the conceptual approach, because no abstract concepts have been introduced.
I.e. the conceptual approach does contain the other, but not vice versa.
