Do I have to use a secondary antibody that is specific to the isotype of my primary antibody? For example, if my primary is IgG2a isotype, do I have to use anti-IgG2a to detect it?

No, antibodies designated as anti-IgG (H+L) with no subtype specified will cross react with all sub-types of IgG, as well as other isotypes like IgM, because they react with epitopes on the both the heavy (constant) and light (variable) regions of the antibody.