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This question is rather vague, but it sounds like you enjoy experiential travel, where you visit a place and try to involve yourself in local experiences. For me, that tends to produce greater lasting memories than looking at a museum or attraction.
Sometimes, you can get this by planning a trip around local special events. Some such events may be primarily aimed at tourists, while others are local affairs. Examples of this kind of thing include:
You can find guides to such events around the world. Travel and Leisure has a Festivals + Events page as does Lonely Planet and Rick Steves for Europe.
It's also possible to infuse some experiences into a more typical trip to mix up the blocks of "shrine or something else boring" time with something more fun:
That said, as experiential travel is gone from something people do to a marketing buzzword, it's worth keeping in mind the advice of NomadicMattβExperiential Travel: Just Another Way for People to Take Your Money. In short, be leery of packaged experiences, but do try to make your own.