Oftentimes when considering certain vacation destinations, I feel that two separate trips are necessary. One to do all the touristy things, see the "required viewings" and whatnot, and one to do the more laid-back, scenery small-country type stuff. Sure you could do both in one trip, but for typical American vacations, where it's usually a week or less, in most cases I'd try to do one or the other on a single trip.
Personally, I wouldn't be actively trying to "not seem like a tourist" but I think some prior knowledge of customs is a good thing. In some places, tipping for your meal can be viewed as an insult. Now, I'm sure in many of these places it's not necessarily viewed as one when it comes from an American, since it's typical behavior for them, but it's still something I would make myself aware of before visiting someplace else. This type of information can oftentimes be found in such guides as those with a title of "how to not seem like a tourist".
It's like when people come to my state and hit the big tourist spots for seafood, or the tourist guide suggestions for good Italian food. Sure those places are alright, but as a native, I don't consider them the best experiences for those kinds of food. In the souvenir shop I work, it's not uncommon to have people ask where the best restaurants are for an "authentic" experience. There's nothing wrong with that; some people want to experience local spots that aren't specifically marketed to tourists.