I have the vector bundle $E:= \mathcal{O}(1)^{\oplus k}$ on the projective line, $\mathbb{CP}^1$. I want to know what are the possible subbundles of this, and why?
We know that the Birkhoff-Grothendieck Theorem says that any bundle on $\mathbb{P}^1$ must be a direct sum of line bundles. So any subbundle of $E$ must look like $\oplus_i^r \mathcal{O}(a_i) $ (where $r\leq k$). So what are the constraints on the $a_i$? Naively, I'd like to say that the $a_i$ just have to be 1 to match the decomposition of $E$, but I don't think this is true.
And the same question, but more general would be what are the subbundles of $\oplus_i^k \mathcal{O}(b_i) $. This would probably be more useful to more people.
While the general question can be answered, the answer is not so nice, so let me concentrate on the first case of $E=\mathcal{O}(1)^k$. As you said, $r\leq k$ and if $r=k$, $E$ is the only subbundle. So, assume that $r<k$. Then of course, $a_i\leq 1$ and this is the only constraint. That is, any $\oplus_{i=1}^r \mathcal{O}(a_i)$ with $r<k, a_i\leq 1$ can be realized as a subbundle of $E$. The proof is essentially an application of Serre's theorem (and induction on $r$) which says that a globally generated vector bundle of rank greater than the dimension of the variety has a nowhere vanishing section. You can see a proof in Mumford's book on Curves on a surface, for example.