Student or Learner
"The Doctor asked me to get few blood tests done one week after I have started taking the new medicines that he has prescribed."
"The Doctor asked me to get few blood tests done one week after starting the new medicines."
I am confused where to put full stop in this sentence?
In BrE writing, you have put the full stop in the correct place - inside the quotation marks. BrE and AmE differ on some punctuation rules.
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.