Will I have any side effects from taking penicillin if I don't have anything?

All drugs have side effects, but if you aren't allergic to penicillin, they ought to be minimal or not noticeable. Because of the concern about allergies, I'm a bit surprised you were given penicillin without a diagnosis. Or that you'd be taking it orally - the usual treatment for this STD is a single injection. If you have a contact with syphilis, however, it makes sense that you'd get antibiotics before getting test results.