Tangent bundle of the 2-sphere

I'm reading through Tu's Introduction to manifolds and today I learned about tangent bundles and vector bundles. I was surprised to learn that $TS^2$, tangent bundle of the 2-sphere, isn't trivial (i.e. $TS^2 \not \simeq S^2 \times \mathbb R^2$. I learned that it could be seen as a corollary to the Hairy ball theorem.

I know the question is extremely vague, but how does $TS^2$ look like then? What do we know about it's topological/differential properties? Is there some way we can visualize it?

What are the other ways to see the non-triviality of $TS^2$?

I know I'm little out of my depth here and that I probably won't understand all the answers, but I hope they could motivate me to learn more of the differential geometry. Also, it's always good to get a little taste of what's ahead before you see your first definition.

As a corollary of the hairy ball theorem, $S^2$ is not a parallelizable manifold, that is, it does not have a set of $2$ globally non-vanishing vector fields that span its tangent space at every point. The parallelizable vector fields can be used to introduce a trivialization.

Maybe a nice excersise to help visualizing the tangent spaces of the spheres is the following:

$TS^n=S^n\times S^n - Δ$

where $Δ$ is the diagonal $Δ=\{(x,x)|(x,x) \in S^n\times S^n \}$.

To see that, imagine what would hapen if you tried to close up each tangent space to a sphere minus a point, using the stereographic projection.

(For a more explicit explanation as to what $TS^2$ is and what is not, see the introduction of Moore's lecture notes on Seiberg-Witten theory. Beware though that he uses quite a lot of things,like the $H^*(S^2)$, to characterise the tangent bundle)