TITLE: concrete examples of tangent bundles of smooth manifolds for standard spaces
QUESTION [28 upvotes]: 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 $T(\mathbb{T}^n)\approx\mathbb{T}^n\!\times\!\mathbb{R}^n$.

REPLY [8 votes]: Here is a concrete perspective on tangent bundles that might help. Take an embedding of your manifold $M$ into $\mathbb{R}^N$ (by Whitney we can always do this). This gives an embedding of $TM$ into $T\mathbb{R}^N = \mathbb{R}^N\times \mathbb{R}^N$ (it's injective and you can write down the differential to check it is an immersion). This already gives you your tangent space inside a manifold with one chart.
Suppose that $M = f^{-1}(0)$ where $f:\mathbb{R}^N\rightarrow \mathbb{R}^k$ is a 
smooth function and 0 is a regular value. Then $M$ is a $N-k$ dimensional submanifold of $\mathbb{R}^N$ and $TM \subset T\mathbb{R}^N$ is $Df^{-1}(0,0)$.
For example, consider the function $f(x_1,\ldots ,x_n) = x_1^2+\cdots +x_n^2$. 1 is a regular value and $f^{-1}(1)$ is the $n-1$-sphere with its standard embedding. The derivative of $f$ at $(x_1,\ldots ,x_n)$ is the $1\times n$ matrix 
$Df_x = 2( x_1 \, x_2\, \cdots \, x_n)$ and the kernel of this map is the 
set $y\in \mathbb{R}^n$ such that $y\cdot x = 0$ (standard inner product).
Therefore, we have an identification of $TS^{n-1}$ with 
$$\{(x,y) \in \mathbb{R}^{2n} | x\cdot y = 0, |x| = 1 \}.$$
The advantage to this perspective is that things are computable now (such as geometric structures). The disadvantage is that this is computationally practical only if you have a simple embedding into euclidean space, the description is not intrinsic (although the tangent bundle is), and it does not say nice abstract things about the tangent bundle.