I'm having trouble visualizing what the topology/atlas of a tangent bundle $TM$ looks like, for a smooth manifold $M$. I know that $$\dim(TM)=2\dim(M).$$

Do the tangent bundles of the following spaces have any "known form", i.e. can be constructed (up to diffeomorphism) from known spaces $\mathbb{R}^n$, $\mathbb{S}^n$, $\mathbb{P}^n$, $\mathbb{T}^n$ via operations $\times$, $\#$, $\coprod$?

- $T(\mathbb{S}^2)=?$
- $T(\mathbb{T}^2)=?$
- $T(\mathbb{T}^2\#T^2)=?$
- $T(k\mathbb{T}^2)=?$, $\;\;\;k\in\mathbb{N}$ ($k$-fold connected sum $\#$)
- $T(\mathbb{P}^2)=?$
- $T(\mathbb{P}^2\#\mathbb{P}^2)=?$
- $T(k\mathbb{P}^2)=?$, $\;\;\;k\in\mathbb{N}$ ($k$-fold connected sum $\#$)
- $T(\mathbb{S}^n)=?$
- $T(\mathbb{T}^n)=?$
- $T(\mathbb{P}^n)=?$

($\mathbb{S}^n$ ... n-sphere, $\mathbb{T}^n$ ... $n$-torus $\mathbb{S}^1\times\ldots\times\mathbb{S}^1$, $\mathbb{P}^n$ ... real projective $n$-space, $\#$ ... connected sum)

I'm making these examples up, so if there are more illustrative ones, please explain those.

BTW, I know that $T(\mathbb{S}^1)=\mathbb{S}^1\times\mathbb{R}$ by visually thinking about it.

P.S. I'm just learning about these notions...

ADDITION: I just realized that all Lie groups have trivial tangent bundle, so $T(\mathbb{T}^n)\approx\mathbb{T}^n\!\times\!\mathbb{R}^n$.