Intuition on the higher order tangent bundles

Question

The first order tangent bundle $TM$ can be thought of as the set of velocities at each point on the manifold.

It can be formally defined in one of two ways: it can be the set of derivations on the manifold or the set of equivalence classes of tangent curves.

I’ve read that the tangent bundle is itself a smooth manifold so you can then take the tangent bundle of that.

My question would be how that would work. When I think of the tangent bundle, I think of a bunch of tangent spaces and I can’t grasp how they make a smooth manifold. Given the smooth structure, how would a higher order tangent space be defined and what would it mean? I’ve seen $T^2 M$ be defined as the set of second order derivations, as equivalence classes of curves that agree to to their second derivative, and recursively as $T^k M \equiv T(T^{k-1} M)$.

How are they related, what would the dimensionality of these tangent bundles be, and what is the geometric intuition behind it?

1. How is the tangent bundle a smooth manifold?

2. What is the definition of higher order tangent bundles?

3. What is the geometric interpretation of them?

4. What math can be done with them?

5. Bonus: Could you explain the same for the cotangent bundle and its higher analogues along with mixed bundles like $T^6 T^{3*} M$?