Since geometry and algebra are often discovered to be two sides to the same phenomena, I suggest that you develop your geometric intuition to understand algebraic phenomena. A typical example here is to use the abstract tensor formalism to understand Hopf algebras. From my personal experience, Hopf algebras did not come alive until I understood that the axioms could be drawn as little bits of string. When I listen to a Hopf algebra talk now, I try to envision the proof via these diagrams.
This leads me to a second point which no one yet has suggested. knot theory is an inherently visual subject that is easily entered via geometric intuition.
To truly make progress as a research mathematician, you may have to also develop tools for symbol manipulation. Geometry can always be a guide to discovering the formulas. Knot theory, abstract tensors, linear algebra, and group theory all are easily approached via geometric techniques. Many of us delight when an arcane algebraic concept is reinterpreted as a geometric one.
