My personal preferred way to explore a new city is to take a hop-on-hop-off bus tour the first day. If you have a 48-hour ticket, you can ride around once to get an overview, then ride again with stop-overs at the places you found interesting.

Another idea, particularly if you want to meet some locals, is to check Eventbrite and Meetup for the city you plan to visit. I’ve found a multitude of happenings that way, mostly free.
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