Hello,
Since you had unprotected intercourse, so firstly you should clarify the status of pregnancy and then start with birth control pills. For the same, you can get a serum HCG also done which is more specific than urine pregnancy test.
If pregnancy is ruled out, you can use any combined oral contraceptive pills. for birth control pills, generally you are to start them on the first day of menstrual cycle or first Sunday after menstrual cycles and you’re given a pack that usually contains 21 pills, and you take one every day for three weeks. At the end of those three weeks, you break for a week. During those seven days, you'll have your period.
It is very difficult to precisely confirm a diagnosis without examination and investigations and the answer is based on the medical information provided. For exact diagnosis, you are requested to consult your doctor. I sincerely hope that helps. Take care and please do keep me posted on how you are doing.
No, the pill is not enough to stop pregnancy. If you had not quit a month ago and had rather quit when you finished a pack, then started your period, and started the pill again on time, you'd be fine, but this is not the case. You will need Plan B to prevent it in case you ovulate within the next five days.