There is no public transport into any Everglades entrance. If you are not driving, you have three options and one non-option.
Organised day trips are the default, and the Miami market has hundreds of them. Most bundle an airboat ride on tribal or private land with a wildlife show and, sometimes, a stop inside the park. Read the itinerary for the words "national park" and check which gate; plenty never cross the boundary.
Rideshare will get you to Shark Valley from downtown in about 45 minutes, and then you are standing at a visitor centre with no reliable way back. Drivers do not cruise the Tamiami Trail. If you try this, pre-arrange the return.
A rental car for one day is usually the cheapest and easiest answer if there are two or more of you, and it is the only way to reach Flamingo on your own schedule.
The non-option is cycling from the city. The Tamiami Trail is a two-lane highway with heavy truck traffic and no shoulder worth the name. Bring or rent a bike at Shark Valley for the loop itself instead — that ride, on a flat paved road closed to private cars, is one of the best things you can do in South Florida.