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Half the fun of traveling is being somewhere else. Don't limit yourself, get out and stretch your legs. Cover new ground with your eyes and ears open.
One of my favorite tricks is to take a cab or a bus somewhere and then walk back. Engage the driver and learn about your destination. Remember: it's all about the neighborhoods. Take a camera and really make an effort to use it. Use it as a tool to help you see not just look at things.
When we traveled with young kids we gravitated towards school yards and nice city parks. Both are often visible on maps. Or ask. Again, it gets you into a neighborhood. Kids meet other kids no matter where they are. We've also bought lunch and headed to a park to eat it.
Don't laugh: take the bus. It's local by nature, immerses one with the natives, and usually goes from point A and point B, both of which have been deemed by public officials to be worthy of a designated bus route. Who are we the traveler to question such wisdom? Ride the bus, subway, and ferry. Getting there is half the fun.
Ask the locals where they go. Start with the hotel staff and/or restaurant staff. Don't laugh, it seems cliched, but believe me I would have never found that Salvadoran restaurant in the Mission District in SF or the better walk from the hotel to Goose Island Brewing in Chicago if not for the hotel staff.
Surf the local alternative weekly newspaper's web site before you go. Look for clusters of art galleries, interesting restaurants, music stores, bookstores, etc. Such clusters represent the neighborhoods you'll want to explore.
How can you leave town without checking out the local museums? They're guaranteed to be different than what you have at home. Often closed on Monday's but low cost other days of the week. Don't forget the small local museums including historical societies.
When in Rome... I find it a kick to listen to local weather forecasts and traffic reports. It assumes a shared geography complete with nicknames that you're not privy to.