Hi K. -
I agree with the other mom who responded, get involved in a mom's group that has children around your son's age. When I first moved to Tampa, I had the same dilemma...what in the world is there to do that doesn't cost money? You're right, the summer months are unbearable so going to the playgrounds is a real drag. I do, however, recommend city pools during the summer because they're free and fun for kids. They often offer swimming lessons for nominal costs.
My husband takes my four-year old son to the Gandy Pier to fish...I pack them a cooler full of snacks, and they're set for hours. My husband and his cousin also take their boys out on their little boat from time to time. My son LOVES to explore old looking houses and buildings (a guy after my own heart!), so I drive around S. Tampa and St. Pete looking for historic homes and buildings. I take pictures of them and we go home and try to figure out the story behind each building. One of our favorite places to explore history is downtown Tampa, particularly the University of Tampa. Now that it's getting cooler, we're going to picnic by the Hillsborough River.
Some other activities that are great fun for kids, but not free are trips to the zoo and Busch Gardens. I like the annual passes because we definitely get our money's worth...my kids LOVES animals. Britton Plaza in S. Tampa has Wacky Wednesday movies where kids' movies are real cheap...maybe you could check that out.
Sometimes, I just like to chill out with my kids at home watching a good cartoon DVD, eating popcorn, and snuggling under covers. These days with gas being so high, anytime I don't have to run around all over town is a treat for me. :)
Also, moms shouldn't feel pressured to ALWAYS seek out interactive, enriching, and educational things to do for kids...letting them explore things in their own backyard and encouraging their imagination is just as important as putting stimuli in front of their faces.
Blessings to you and yours.