Living in Florida one gets a lot of opportunities to see the very unique environment of the southern tip of the country. The Everglades are truly unique, and easy to access. While some tour companies do exist, I recommend a car for this. Go at your own pace, take as much time as you want. The pace of life is slow, so enjoy it by yourself or with your loved ones.
The high season in Southern Florida is never. Sure you get the old snow birds who come down for the fall/winter, the spring breaking families who come down in spring and the Florida locals who visit in Summer. Thing is, when you start getting to the edges of the park, the landscape spreads right out, any you become small enough to eat. Only a measureable distance from a prehistoric beast, one who has a taste for people. You don't need to outrun the gators, just be able to outrun one or two people you are traveling with. For extra protection bring the whole family!
A trip through Everglades National Park can be an easy day trip from Miami or Fort Lauderdale. Just go to any of the four visitors centers will be the best Park entrances. The National Park Service has that info.
At Shark Valley you can rent bikes and ride around the 15 mile loop. It is a very long bike ride that will take a novice biker several hours. And don't forget the SPF 45. Especially on your knees. Go during the week when you have less families clogging up the road. And yes, there will be times when you hear a gator growl. And yes that is a very unnerving sound.
It is an amazing National Park. Here at the Anhinga Trail we have a mile long walk which guarantees you will see dozens of alligators and migratory birds. Even this **chap to the right must be looking at something so exciting it requires binoculars. **Note: man with binoculars is not an actor.
At the end of Anhinga Trail the payoff includes a couple dozen gator's resting in one apparent orgy of gator goodness.
I guess the "Do Not Swim" sign would have been redundant?
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