A standard reference is do Carmo. However, general tensors are not thorougly approached as far as I remember: he just mentions it en passant.
Another good reference, but that may not be appropriate since you are not acquainted with Riemannian Geometry, is Helgason. Differently from do Carmo, he treats tensors extensively.
Now, tangent bundle and tangent space are not objects exclusive to Riemannian Geometry. You can find good explanations about those objects in Bredon, which is a remarkably excellent book and also in Bröcker, Jänich. One reference for this is also Hirsch, but Hirsch, as Helgason, may be too thorough for a first approach. Furthermore, Hirsch restricts himself to a specific definition of tangent vectors/tangent space which is useful, easily generalizable, intuitive but does not communicate very well with Riemannian Geometry, in which we like to visualize tangent vectors as derivations a lot of times. In that respect, the treatment of Bröcker, Jänich is very appropriate: he presents three definitions of tangent space.
I hope I was of some help.