The Old Town

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Your destination is Casco Viejo, Panama City's old town with a variety of lodging options depending on your budget and taste, from a hostal bunk at Luna's Castle to marbled baths and Italian linens.
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With your bags dropped, head out for a stroll around the neighborhood. Casco Maps are available at Super Gourmet, along with a friendly-vibe and some good coffee, snacks and local products. Don't miss the National Theater, the La Merced church, the San Jose church and its golden altar, the Arco Chaco, the Plaza de Francia, and the old fortress wall at the point, where artisans set up shop daily. Other good options for souvenirs are the shops along Calle 1era. 
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Stop in the Canal Museum on Plaza Catedral, which has a wealth of canal history for a $2 entrance fee.
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If you are traveling on a budget, stock up on snacks and drinks at one of the local corner stores, more commonly known as Chinos, because most are owned by Chinese. Papo's is also a great option for lunch and local flavor - affordable plates of grilled chicken accompanied with your choice of sides for under $5.
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Grab a coffee and bagel at Casa Sucre Coffeehouse or a nice cold Panama or Balboa beer at one of the bars at the entrance to Casco Viejo near Luna's Castle, great for an afternoon happy hour and people watching.
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For dinner, try the fresh fish tacos and craft beer at The Fish Market on Avenida Central or the authentic Italian pizza at Caffe per Due. If you are feeling more adventurous, there are a number of local fondas for cheap eats.
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And for the evening, you couldn't be better placed for rooftop cocktails, plaza people watching, bar hopping and DJ parties.