Should I stop taking finasteride & minoxidil before/after FUE hair transplant? (Photos)

Given your family history I would not recommend a hair transplant until your late 20's. Stay on finasteride and minoxidil and if your hairline does not recede dramatically go for surgery at this point. Surgery is not an option despite what some surgeons may say. Read about other treatments such as PRP in the meantime if you are desperate for some extra thickness. 

Thank you for submitting your question, at your age, would recommend to continue your minoxidil and finasteride. PRP, laser cap and taking amino acids is recommended. When you are ready for your FUE, you can continue finsasteride, and discontinue rogaine for 1-2 weeks before your surgery.  Best wishes.

It is best to discuss medication related issues with your doctor that will be treating you.  Your doctor can better guide you in the treatment plan.

Generally Finasteride you can keep using immediately before and after the procedure. For Minoxidil, most doctors have you stop few days before and then continue using it around two weeks afterwards. The exact timeline really depends on your doctor's preferences since each has a slightly different pre- and post-op routine.
On another note 24 is very early for a transplant, generally I do not recommend getting a transplant before 26 unless there are special circumstances, such as a scar needing covering, etc. If you continue losing hair it may make the transplant look weird since you'll have full hair along your hairline with thin/bald areas behind it, so you'll be forced to have another. If you don't have enough donor hairs left for that second procedure, you may be out of luck and stuck with weird looking coverage. Definitely consult with one or several doctors in person about this if you haven't already.

If you have already chosen your hair restoration surgeon, then these are questions you should be discussing with him in preparation for you upcoming procedure. 

You are a bit young to have a hair transplant as you will not know, for sure, what will be your eventual hair loss pattern.  When you eventually get a hair transplant, stay on the drugs