Let $E$ a rank $r\geq 3$ vector bundle over a curve $C$ and let $E'$ a rank $r-1$ subbundle of $E$. Thus we have $\mathbb{P} (E') \subset \mathbb{P} (E)$; what can be said about $
\xi|\_{\mathbb{P}(E')}$, where $\xi$ is the tautological class of $\mathbb P (E)$?


If $\xi'$, $F'$ are resp. the tautological class of $\mathbb P (E')$ and the class of a fibre in $\mathbb P (E')$, is it possible to compute the components of $
\xi|\_{\mathbb{P}(E')}$ with respetc to $\xi'$, $F'$?



Restriction of the tautological class to a subbundle
I personally prefer to work with sheaves which ultimately gives you the same thing, but sometimes you need to work in a dual setting. So, I will use sheaves below, feel free to rewrite this for yourself in the language of bundles.


Let $\alpha: \mathscr E\_1\to \mathscr E\_2$ be a surjective morphism of sheaves and a scheme $X$ and let $\pi\_i:\mathbb P(\mathscr E\_i)\to X$ be the associated projective bundle for $i=1,2$.


By construction there exist surjective sheaf morphisms $\pi\_i^\*\mathscr E\_i\to \mathscr O\_{\mathbb P(\mathscr E\_i)}(1)$ on $\mathbb P(\mathscr E\_i)$. Pulling back $\alpha$ to $\mathbb P(\mathscr E\_2)$ gives a composition of surjective morphisms:
$$
\pi\_2^\*\mathscr E\_1\to \pi\_2^\*\mathscr E\_2\to \mathscr O\_{\mathbb P(\mathscr E\_2)}(1).
$$


This induces a natural map $\beta: \mathbb P(\mathscr E\_2)\to \mathbb P (\mathscr E\_1)$ such that $\beta^\* \mathscr O\_{\mathbb P(\mathscr E\_2)}(1) = \mathscr O\_{\mathbb P(\mathscr E\_2)}(1)$. I leave it for you to check that $\beta$ is injective (after all you claimed this as a matter-of-fact in the question). The pull-back equality means that the tautological class of $\pi\_1$ restricts to the tautological class of $\pi\_2$.


In other words, using your notation, $\xi|\_{\mathbb P(E')}=\xi'$.


