There is no border between Israel and the West Bank. Once you are in Israel you are free (as a tourist) to go wherever you like in the West Bank.
Thousands of tourists visit Bethlehem every day, and this raises no eyebrows whatsoever. Other West Bank cities are less visited by tourists, for fairly obvious reasons, but going to Ramallah or Nablus just to have a look round, sample the food, etc., is a totally normal thing for tourists to do and nobody will have any objection. There will be no evidence in your passport that you did this. (But take your passport when you go; you might need to show it at roadblocks to prove you are a tourist. You will almost certainly be waved straight through.)
Israel no longer stamps passports at Ben-Gurion Airport, so assuming you enter and leave by plane your passport will have no evidence of your visit to Israel.
If airport security personnel ask you at the airport, when you leave, if you visited the West Bank, answer truthfully. Don't make political statements; just say you were visiting as a tourist (assuming that's the case).