To avoid writing a long essay I'll be very telegraphic.
Here are some of the many open problems that are reasonably "next" in the area. I'll use [GLC] to refer to the recent papers proving the unramified geometric Langlands conjecture over the complex numbers.
I'll start in the unramified context over C, then leave the unramified setting, and finally leave C.
Prove GL with other sheaf theories: [GLC] proves the de Rham, Betti and étale [aka restricted] versions over C. What about the Dolbeault [aka "semiclassical"] version of Donagi-Pantev? AFAIK a complete formulation (a la Arinkin-Gaitsgory) doesn't yet exist but a lot is known. (Physically this concerns a limit which is no longer a TQFT). It is also natural to ask for a version for motivic sheaves interpolating all of these, in the spirit of Scholze's current course. (Physically this concerns "1/2 BPS branes").
Quantum GL: the GLC (and underlying physics) has a one-parameter deformation involving quantum groups (Betti) or noncritical level reps of affine Kac-Moody algebras (de Rham). At roots of unity this relates to metaplectic phenomena in the Langlands program.
Relative GLC. The relative Langlands program in this context predicts a large collection of objects on the two sides of the correspondence that should match. Some matchings are crucially used in [GLC] but in general this is wide open. Physically this corresponds to considering boundary conditions in the TQFT.
I think those are the main open problems in the unramified context -- ie evaluating the 4d TQFT on closed surfaces. The most obvious "next" is allow ramification. In the case of tame ramification (Iwahori level structure / parabolic versions of everything) this is easy to formulate. To allow general ramification, one needs to grapple with:
- local geometric Langlands - a spectral theory of the 2-category of categorical representations of loop groups. This is physically the duality evaluated on a circle.
Given matching objects under local GLC one can formulate global GLC with ramification (ie the TQFT on 2-manifolds with boundary).
Or one can "go up" -- study the TQFT on 3-manifolds. The proof of [GLC] implies that there are well defined 3-manifold invariants around (since the Betti spectral side is a TQFT), so one can try to formulate and prove Langlands for 3-manifolds (cf eg a recent survey of David Jordan).
A closely related direction is the analytic Langlands of Etingof-Frenkel-Kazhdan, which (a la Gaiotto-Witten) is the duality on certain 3-manifolds w boundary.
Next in proximity is GLC on the twistor P^1 as developed by Scholze, which is the strongest known form of the real local Langlands program. The "regular" part of this was formulated by Nadler and me as an aspect of the physics duality evaluated on UNORIENTED 2-manifolds.
Now we leave the complex numbers. Arguably the most obvious "next" to think about (and I'd bet what the authors of [GLC] are thinking about) is the étale unramified GLC in positive characteristic. This was formulated by many of the same authors (plus Kazhdan and Varshavsky) and shown to imply [a very strong form of] the unramified Langlands correspondence over finite fields.
Of course one can then try to do the tamely ramified story there, and dream of more general ramification.
Then there's etale GLC on the Fargues-Fontaine curve a la Fargues-Scholze, the strongest known form of the local Langlands program, which is again a form of unramified GLC but in a more exotic setting. This has forms with various coefficients - for example there's an emerging de Rham form which gives a strong form of p-adic local Langlands (with the spectral side provided by the Emerton-Gee stack).
One exciting thing about the de Rham p-adic Langlands version is it's the only other setting where the main techniques of [GLC], which are very special to the de Rham setting (the use of rep theory of affine Lie algebras, opers etc) might apply.
(One can also consider relative Langlands versions of all of these statements in case this all seems too limited in scope.)
And finally there are number fields out there as well...
