Some signs might start showing as early as a week or two along. You can look here
http://www.americanpregnancy.org/gettingpregnant/earlypregnancysymptoms.html to find out what those symptoms are.
They're usually faint and confusing though. I will tell you what I tell almost everyone in here, do not go by symptoms. You can never determine whether or not you are pregnant by the symptoms alone. Early pregnancy symptoms are vague and the same as the symptoms for many other things including stress, worrying if you are pregnant, and an oncoming period just to name a few. The only way to know is to test.
Also, not all women have the same symptoms. Implantation bleeding, for example, is an early sign, but only about a third of women experience it.
Yes, it is true that sperm can live inside the v****a for a few days. That means if you have sex on Sunday and ovulate on Tuesday, for example, the sperm might still be alive and well in there to fertilize that egg.
Keep in mind that you can get pregnant at any time of the month, even when you have your period.