If it makes you feel any better, this would take up at least one two-hour lecture in a decent graduate-level university course on GR. If you are going to try a

If it makes you feel any better, this would take up at least one two-hour lecture in a decent graduate-level university course on GR. If you are going to try and comprehend it on you own, I am personally fond of Simmonds' "A brief on tensor analysis" as an introduction to the way vectors are understood here.