We used to live close to Fort Myers and spent quite a lot of time in the area and surroundings. This was quite a few years ago before the hurricane devastated the area.
There is Fort Myers as well as Fort Myers Beach.
You might like to look at Youtube showing you the devastation and hear about the updates on the work. They cleared some of the beaches, that I know, but I would not be able to tell you where to stay at this point. We used to stay on Fort Myers Beach, Sanibel Island,or Captiva Island. They had a beautiful resort there that offered all kind of sports and water activities.
I know that they let go of all their staff and need to rebuild entirely. These islands had a lot of charm and represented old Florida nicely.
For sports, I would recommend Marco Island as a destination.(not too far from Naples). Both of those cities also had damage but not as intense.
Naples is a beautiful city to visit and do some shopping or just browse.
You can do quite a lot around that area.
For shopping there is the big Miromar Outlet Mall and lots of small boutiques in each city mentioned above
To eat there have a lot of fresh seafood and great restaurants.
(I do not know which ones are open at this point. You might like to use Yelp to guide you on this and call the restaurant before going there to make sure they are open.)
To visit: the boardwalk trail at Six Mile Cypress Slough Preserve, the Sanibel Island Wildlife Refuge.
Paddling, kayaking, and fishing are big in the area as well as golfing and tennis. Perhaps more pickleball now?
Some big sport teams do their training in the area as well. (Not sure if they are present in June)
The Edison House is in Fort Myers to be visited.
In Naples, you should go to the Audubon's Corkscrew Swamp Sanctuary. Well worth seeing.
They have also guided tours through the Babcock Ranch Eco Tours.
Finally, the area has a lot of beautiful state parks to visit. Check the Florida State Parks website for more info on each of them.
As I mentioned above, make sure to contact the Chamber of Commerce to see if all the places are in operation.
Be ready for high temperatures in June. It will also be very humid.
Fort Myers and Fort Myers Beach have quite a lot of traffic. The airport is very accessible and I-75 will take you wherever you want to go!
Have fun and enjoy beautiful Florida!