How should I think about this subset of a product of projective lines? Consider the group $G = \mathrm{GL}_2(\mathbb{C})$, and it maximal torus $T$ of invertible diagonal matrices. The coset space $G/T$ should have the structure of a (smooth?) variety/scheme/manifold. Direct computation shows that $G/T$ can be described as the space
$$ G/T = \{([a:b], [c:d]) \in \mathbb{P}^1 \times \mathbb{P}^1 \mid ad - bc \neq 0\}$$
How should I think about this space? For example, there seem to be many line bundles over this space that have an infinite-dimensional space of sections, as I found by attempting to induce characters from $T$ up to $G$, but I'm not sure of the reason for this geometrically. I'm also having trouble with some basic questions: is it connected? And importantly, where can I learn the theory required to understand this? (At least at an "intuitive" level).
 A: Here is a more detailed answer : there is a natural projection map $p : G/T \to \Bbb P^1$, projecting onto the first component. The fiber over $(a:b)$ is $\Bbb P^1 \backslash \{(a:b)\}$. 
Such map has a nice interpretation : if $B$ is the group of upper-triangular matrices, $G/B$ can be interpreted as the complete flag variety in $\Bbb C^2$ i.e $\Bbb P^1$. The canonical map $G/T \to G/B \cong \Bbb P^1$ exhibits $G/T$ as a $B/T \cong \Bbb C$-bundle over $G/B \cong \Bbb P^1$.
This automatically shows you that $G/T$ is connected. Alternatively one could also embbed $Q = \Bbb P^1 \times \Bbb P^1$ in $\Bbb P^3$ as a quadric surface via the Segre embedding, and $G/T$ should correspond to the complement of an hyperplane section of $Q$ which is connected.
Now the tricky part comes, computing which bundle is $G/T \to G/B$. My guess is a correct approch would use something like Borel-Weil theorem (in fact the chapter about homogenous spaces in "Representation theory, a first course" by Fulton and Harris looks very nice, maybe it's a good place to looks even if they essentially look at $G/B$ for $B$ a Borel subgroup) and identify it nicely. 
Since any bundle over $\Bbb P^1$ is $\mathcal O(k)$ we just need to check how many zeroes/poles has a generic section. I pick $\sigma : G/B \to G/T, (a,b) \mapsto ((a,b);(-b,a))$. This map is a rational section, with only two poles at $(1, \pm i)$. This means that $G/T \to G/B$ is the cotangent bundle of $\Bbb P^1$, namely $\mathcal O(-2)$.
