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Ecuador is by far on my top five list of favorite places. Not only is it home to the equatorial line that divides the northern and southern hemisphere, it also is home to the Amazon Rainforest and my personal favorite...the Galápagos Islands!
I would advise flying into the capital, Quito, and staying there a few nights to explore its culture and history. Quito is the highest capital in the world (2,850 meters above sea level). Definitely take the cable car to the top of Pichincha Volcano. You can catch the 'teleferico' at the station in the heart of Quito to see amazing views of the entire city. I'd recommend strolling through Old Town to see the city's historical buildings and to try some delicious, traditional street food.
When you're ready to leave the city life, stay a few nights in the Amazon. Opt for the primitive lodging and sleep in hammocks...trust me, it's worth it. Most Amazonian "hotels" offer activities such as a nighttime jungle hike, an early morning bird watching expedition or a trek to a native village.
After exploring a good portion of mainland Ecuador, fly on over to the Galápagos Islands. There are two islands that you can fly into, Baltra Island (claiming to be the world's most ecological airport) and Puerto Baquerizo Moreno on San Cristóbal Island. From there you will take a ferry to (I recommend) the port town of Santa Cruz.
Walking around Santa Cruz was spectacular. All the buildings are very colorful, definitely giving off an 'island vibe'. The fish market is also really fun to watch because you're basically competing with the sea lions to get the fish first. The sea lions hide under the tables of the market waiting for the moment that they can jump up and scoop their next meal.
The Galápagos are home to the Charles Darwin Research Center where you can see the infamous, gigantic Galápagos Tortoises. I spent a day exploring the center, learning about wildlife conservation and the lives of the tortoises.
Another must is taking a snorkeling trip out to San Cristóbal Island and not only snorkeling, but also swimming with sea lions!
Caution: Though sea lions seem playful, do not touch them or get too close to them. They will bite!