It depends on the city/area. In some places it's allowed, in others it is not.
Then their shop isn't certified as kosher and can't make food for religious customers, but otherwise, it works fine. I've gone to convenience stores, cafes, restaurants, movie theaters, etc dozens of times on Shabbat, in Israel.
Were you able to take the trains? Buses?
Yes, of course. I lived in Haifa, which kept its bus rapid-transit system and several bus lines running all through Shabbat. This was especially helpful for the nighttime bus lines, so I could go to the beach on Friday afternoon, eat dinner and see a movie at Lev haMifratz with friends in the evening, and then get home at maybe 1-2am, all with public transit.