How can one visualize topological quotients or develop intuition for handling them?

There is a bit of case-by-case here, but when the quotient is "nice" enough, you can often use a fundamental domain to visualise it.

You probably know how to construct a torus as a quotient of the plane by identifying points that are translations apart. You can visualise this torus also by taking the parallelogram generated by the two translations and thinking like pac-man. That is, following a point inside the region and if it hits one of the sides, it pops back on the other side since the two sides have been "identified".

This also works for the projective plane quotient you talked about. Since the antipodal points are being identified, the northern hemisphere of the sphere is a fundamental region, that is, it contains exactly one point in each equivalence class. So you can see the projective plane as a hemisphere, where when you move a point to the equator it pops back to the opposite side. Since we're thinking only topologically here, you could also "flatten" this hemisphere to a disk, so the projective plane is like a disk where when you reach the border you reappear at the antipodal point, much like the torus picture.