If you like food, shopping, and old buildings, head to Quincy Market. It's a nice area to walk around if you have some free time. The main building has virtually every type of food you could want (we recommend the pizza bagels). It's also not far from the harbor if you want to walk down by the water.
Located right off of BU's campus, this is where we would go for frozen yogurt, even long before we were dating. They offer dozens of toppings and mix-ins to make your own, which is why we often stood at the counter for several minutes making a decision.
A short walk from the Haymarket stop on the Green Line, The North End is Boston's Italian district. This is where we had our first date (at Bella Vista Restaurant). Head down Hanover St. if you want to grab a cannoli at Mike's Pastry (cash only) or some gelato.
The esplanade is one of the best places to go for a walk, and in October it should be especially beautiful. Definitely make your way there if you're around before or after Sunday. This will also be where you can watch the Head of the Charles on Saturday.
If it's your first time in Boston, this is a good way to see the city all at once. Tours depart from multiple locations, so check the website for more details.