Welcome to the community!
I think the quick story here is that your humoral immunity can be knocked down while until the first couple of days post Rapamycin. So bacterial infections are increased - and should be treated early.
Viral infections in general should be a bit less as your T cell immunity should usually do better, which includes fighting viral infections and identifying malignancies.
That has been my experience on this. If you’ve gotten more colds - it could simply be bad luck.
Did you have any blood testing prior to starting? I would see an immunologist for an immune work up. CBC, IgG, A, M, E; MBL, flow cytometry, specific titers, delayed hypersensitivity, etc. etc.
It’s possible that you had some minor subclinical immunodeficiency like mild neutropenia or common IgA deficiency that was exacerbated by adding a systemic immunosuppressant.
I would definitely would stop rapamycin unless it was prescribed for a specific reason by a physician.