Most of my travel experience has been in short spurts, never staying in any particular city for too long. I travel light as to be flexible with means of transportation.
When I have free time in cities I'm less familiar with I try to avoid the really touristy places, although art galleries, museums and historical landmarks are always interesting. I seek out the coffee shops with the best Americano, comic/book stores, record stores with a good selection of vinyl, vegan friendly places to eat, and then finally if it's winter or spring, somewhere to have a pint and watch the hockey game.
The Internet is making it easier to find the "good spots". You can use Yelp, Urbanspoon, Google Maps and a thousand other websites to read every review and get all the insight you need before ever leaving home. When it comes down to it though you just need to get outside walk around and talk to people. In the good cities one can easily walk or take public transportation to all of the places mentioned above.