I think it will be similar to how they show the Scarlet Witch using her power, with some light effects, and how she seems a little out of synch with the world at times. Of course, on a level much higher, because he is taught, where she is just figuring things as she goes (unless there is a teacher we haven't seen).
I like the idea of him walking between this dimension and others, and how they might show it. That is the one nice thing about the movie being pushed back, they have even more time to perfect the visualizations. Strange's spells are too colorful to miss out on.