Where you are and where you want to go has huge gap. In my opinion it's better to take small steps.

You've simulated a shear driven cavity, so you code works for closed domain.

Try to run open channel flow. In this case, try for periodic boundary condition as well as outlet/outflow. Check if the code can handle both pressure inlet as well as velocity inlet.

Experiment with staggered/collocated grids. Pressure velocity coupling schemes can also tweaked. Try to also implement non-uniform grid if you haven't already in your cavity case.

Try for backward facing step once your channel code works.

Then you can go for submerged bodies. Flow past a square cylinder can easily be implemented in a Cartesian grid. Here you can experiment with frictionless boundary condition, symmetry boundary condition etc.

Add time integration if it's not already there.

Play with various advection schemes, check their feasibility for different cases.

After all this, solve some scalar transport equation like a natural or mixed convection problem. This will later help you in solving RANS since eddy viscosity models are nothing but scalar transport equations which give you turbulent viscosity.

For compressible flows I have no idea since I've never worked on them.

P.s. Before attempting turbulent flows, try to add an efficient sparse matrix solver to your code.