Let $M$ be a geodesically complete connected riemannian manifold.
Let $p \in M$ be a point and $c: \mathbb{R} \to M$ an arbitrary geodesic that doesn't intersect p. Our aim is to find a "nice" map sending points on the curve $c$ to tangent lines at $p$.
Pick a point $q$ on the curve $c$ and connect it with $p$ via a geodesic curve (this is possible by completeness of $M$). Here is the relavent picture:
Every such geodesic corresponds to a unique tangent line through $p$. Ideally we want to have a cannonical choice of such geodesic for every point of $c$. Assuming there is such a choice we could obtain a function $f: c(\mathbb{R}) \to \mathbb{P}T_pM$. where $\mathbb{P}T_pM$ is the tangent projective space or equivalently the fibre at $p$ of the $(1,dim(M))-$grassman bundle over $M$.
Now my question has two parts:
1) Given a initial point $q$ and an initial choice of geodesic connecting $p$ and $q$, is there then a cannonical choice for every other point on $c$? What i have in mind is a choice that "minimizes" in some suitable sense the variation of the geodesics with respect to variation of the initial point $q$. (I'm not so familiar with jacobi fields but i have a strong sense that this is what's missing here).
2) Under what conditions on $M$ is the map $f$ (defined above) smooth (resp. continuous) as a curve in $\mathbb{RP}^{n-1}$ where $n=dim(M)$. By which i mean the composition $f \circ c : \mathbb{R} \to \mathbb{P}T_pM \cong \mathbb{R P}^{n-1}$ is smooth (resp. continuous).
Here's the picture I'd like to have in mind in this context. It might be wrong though, so you better take it with a pint of salt.
Note: The picture is far from being literal. $M$ is drawn as a surface so the fiber should be a projective line while i drew a projective plane.
I do not know what you mean by "nice", but here is a construction. Let $q=c(0)$. Then $c$ lifts to a parameterized straight line $\tilde {c}$ in the tangent space $T_qM$, such that $c(t)=\exp_q(\tilde{c}(t))$. The point $p$ lifts (noncanonically, unless $\exp_q$ is a diffeomorphism) to a point $\tilde{p}\in T_q(M)$. Now, connect $\tilde{p}$ to $\tilde{c}(t)$ by straight line segments in $T_qM$. These segments project to the required curves $a(s,t)$ in $M$ connecting $p$ to $c(t)$. The velocity vectors $$ \frac{\partial }{\partial s} a(s,t)|_{s=0} $$
define the lines in $T_pM$ that you are asking for and, hence the map to $PT_pM$.
Note that the curves $a(s,t)$ above are not geodesics (in general). In order to use geodesics you have to assume that for the geodesics $\gamma(s,t)$ connecting $p$ to $c(t)$, the points $p, c(t)$ are not conjugate (otherwise you cannot make a smooth choice of $\gamma(s,t)$). For instance, if you assume that $M$ has nonpositive curvature, you can use geodesics $\gamma(s,t)$ for the construction. The clean way to do so is to lift $c$ to the tangent space $T_pM$ via the exponential map $\exp_p$ (since it is a covering map by Cartan-Hadamard theorem) and then use straight line segments to connect via the same procedure I described above.