$\mathbf P^1(\mathbb R)$ is a closed curve, so its tangent bundle is trivial. There is not much to be imagined, really, in that case! In general, what kind of answer do you expect for your «what is the tangent bundle of $\mathbf P^n$?». It is the tangent bundle of $\mathbf P^n$...
Regarding your last question: if a group $G$ acts properly discontinuously and differentiably on a manifold $M$, then $M/G$ is a manifold, $G$ acts on $TM$ in a natural way (also properly discontinuously, and morover linearly on fibers), and $(TM)/G$ is "the same thing" as $T(M/G)$.
Note that $\mathbb{R}^n$ (and thus every finite-dimensional vector space) has a canonical connection given by considering the partial derivative operators $\partial_i$ as parallel vector fields (equivalently considering the directional derivative $U,V\mapsto\partial_UV$ as the covariant derivative operator).
Any diffeomorphism allows us to induce a connection from the domain to codomain, so any chart $\varphi:U\to\mathbb{R}^n$ on a manifold $M$ induces a local connection on $U$, which we'll call the coordinate connection, with covariant derivative $\partial$. The costruction you write above appears to be equivalent to the coordinate connection.
The trouble with coordinate connections is that they are not globally defined, and they are not intrinsic, i.e. different coordinate charts with overlapping domains will in general induce a different connection on their common domain. An example of two such charts Cartesian and polar coordinates in $\mathbb{R}^2$: One can verify that these charts induce different coordinate connections on their common domain.
Ideally, we would like to equip a Riemannian manifold with an intrinsic connection, i.e. one that is completely determined by the manifold and metric. Coordinate connections won't do, since we additionally need to specify a preferred coordinate chart. Instead the Levy-Civita connection, which is uniquely determined by the metric, is most frequently used.
Despite the drawbacks, coordinate connections are still useful in computations. It can be shown that the difference between any two connection operators is a $(2,1)$ tensor, so any (affine) connection can be written in a particular chart as $\nabla=\partial+\Gamma$, where $\Gamma$ is such a tensor.
Best Answer
A lift of an injective path $\gamma:I\to X$ is as you suggest a map $\widetilde{\gamma}:I\to TX$ so that $\pi:TX\to X$ satisfies $\pi\circ \widetilde{\gamma}=\gamma$. Actually, such a lifting is equivalent to specifying a tangent vector along each point of the curve $\gamma$, so you can think of this as a section of the bundle $TX$ restricted along $\gamma$ (aka $\gamma^*TX$). Anyway, the point is that such a lift is the smooth choice of a tangent vector at each point to the curve, i.e. a vector field along a curve like one sees in multivariable calculus.
Now, if we have the additional structure of an affine connection $\nabla$ on $TX$, then we get the definition of which vector fields are parallel along $\gamma$, meaning they satisfy a differential equation $\nabla_{\gamma'}X=0$, where $\gamma'$ is the trajectory vector field of $\gamma$. If we start with a loop $\gamma:I\to X$, (such that $\gamma(0)=\gamma(1)$ and let's say without other self-intersection) then we might ask when a lifting $\widetilde{\gamma}:S^1\to TX$ is a loop in the tangent bundle. In particular, this occurs exactly when we can define a vector field on the image of $\gamma$ by $X_{\gamma(t)}=\widetilde{\gamma}(t)$.
On the other hand, if we fix a loop $\gamma:I\to X$ and choose a tangent vector $X_{\gamma(0)}\in T_{\gamma(0)}X$, parallel transport allows us to define $\widetilde{\gamma}(t)$ to be the unique parallel lift of $\gamma$ with initial condition $\widetilde{\gamma}(0)=X_{\gamma(0)}$. Very often, $\widetilde{\gamma}(1)\ne \widetilde{\gamma}(0)$ and as a consequence we do not get a well-defined vector field along the image of $\gamma$. We can however define a linear transformation of $T_{\gamma(0)}X$ by sending $X_{\gamma(0)}\mapsto \widetilde{\gamma}(1)$, where $\widetilde{\gamma}$ is the lifting with initial condition $X_{\gamma(0)}=\widetilde{\gamma}(0)$. In particular, a loop at $x$ defines a map $P_\gamma\in \operatorname{GL}(T_xX)$ by the above procedure for $x=\gamma(0)$. This leads to the notion of the holonomy group of $X$ at $x$: $$\operatorname{Hol}_x=\{P_\gamma:\gamma\:\text{is a loop based at}\:x\}.$$ If $X$ is a Riemannian manifold with metric $g$ and Levi-Civita connection $\nabla$ then the holonomy interacts with the curvature. Intuitively speaking, nonzero holonomy detects curvature. The idea being that curvature causes parallel transport of curves to deviate from the identity on $T_xX$. The following picture from Wikipedia illustrates exactly the idea:
If we start at the north pole, and move due south to the equator in any direction, then due west along the equator and then due north back to the north pole, the nontrivial curvature in the region bounded by our path of travel causes the parallel-translated vector from the north pole to rotate.
As a last comment, there is an interaction with representation theory when the connection $\nabla$ is flat. In this case, the transformation depends only on the homotopy class of the loop $\gamma$ based at $x$. Hence, we get a group homomorphism $\pi_1(X,x)\to \operatorname{GL}(T_xX)$ called the monodromy representation of $\pi_1(X,x)$, and defined by $[\gamma]\mapsto P_{\gamma}$.