Settle into my "neighborhood", discover things for myself, and people watch. I try to stay at least a week, preferably 3 or more, in one place. I explore the neighborhood where I live thoroughly, looking for green markets, take-aways, ordinary shops like pharmacies and convenience stores, etc. I pretend I live there, going to the same place for my morning coffee and afternoon snack, sitting around reading on a neighborhood bench, bringing home a take-away instead of going out for the evening and so forth. This is part of why I never carry a camera. If I visit a city repeatedly, as I have London, I always stay in the same neighborhood, preferably the same B&B. I've never been there for longer than a week or two, but there's a neighborhood that feels like my London "home" to me.
When I go to a "tourist site", I travel by public transit and get off a stop or two before the one I think is right and use my map to find my way. If I find something interesting along the way, I might never actually make it to the famous place. The more likely that is, the better I like the city. This happened all the time in Barcelona. I'd get entrigued by the fancy doors and stained-glass windows and wander off the track entirely.
If I can find a place where the "life" of the place happens, mothers and babies gather, kids come after school, teenagers "haunt" checking each other out, people meet for dates, I can spend hours there just reading, nursing a drink, and watching. There's a square in Barcelona's Gracia section that fit that bill, and most Paris parks do.
One of the best things I ever did was to start a piece of crazy quilting on a trip. I stitched all kinds of things I found onto it as I worked on it along the way, and turned it into a cushion.
DJC