Today, we talk about the linear approximation of a smooth manifold at a point $p \in M$: the tangent space $T_p M$. There are several approaches to define the tangent space, one possible way is to define it in each coordinate patch, and then show the compatibilities in coordinate patch; another way is to characterize its action on a function by directional derivatives, and characterize the tangent space as such.

In class, I followed basically Lee's Ch 3, first two sections. Spivak also did a very good job explaning what is a tangent bundle, and gives an example of why the tangent bundle of a sphere is not trivial (you cannot comb the hair on a sphere).

Also, I suggested reading about examples of smooth manifolds. Lee's Ch 1, Ex 1.30, 1.31, 1.33 are all worth reading.