Check your local library -- not just the one in your area but others in the same library system if your library is part of a larger county system etc. Our library system here has tons of free activities for kids just his age; in fact that seems to be a target group for library activities, such as a story time followed by a related, kid-friendly craft; a visit from an animal handler who brings birds or reptiles or other animals; a "bedtime story" event where kids come in pajamas in the very early evening; and so on. All free! Go online to the library system or go see the children's librarian at your local library.
Scour the bulletin boards at the library and grocery stores and kids' clothing stores -- many playgroups and other kids' activities put up flyers in those places.
Look for free magazines usually at the doors of grocery stores and other places -- we have ones here called "Washington Family" (yours is probably "Florida Family" etc. and they also list activities and events, many free, many low-cost.
If you can afford it: Try a Music Together or Kindermusik class. I have experience with Music Together and it was fantastic for my daughter at just this age; a good, trained teacher helps kids get into participating, and it is a parent and child class so you will be there participating too.
Look at your county or city Parks and Recreation Department and its listing for kids' classes and one-time events for kids. You may be surprised at how very, very cheap these things can be! Look for classes that might be fun for him; and at his age most will be mom-and-me classes, not drop-offs, so he will be able to adjust gradually to being with other kids and listening to other adults while he feels secure because you're nearby. You might just find that he soon is dashing off to play with other kids or do what the adult leader is doing and loses some of his anxiety -- because he'll know he's in a safe, fun place and that you're nearby.
I understand going to the playground in hopes of finding other kids, but really that is so very hit-or-miss, as you already find. You can't be sure that there will be other kids anywhere near his age, or kids who will play OK together. I really would start with the Parks and Rec department for inexpensive classes and the library for kids' events and activities there. Great resources. He will be fine and you and he will have a great time! By the way -- do think ahead about your younger child now. You may not be able to bring him along to these things, so think now about what do to with him when you take your son to these events or classes. It really is healthy for your older son to have time just with you and without his younger brother there; your older son could end up either relying on his younger brother to stick with him at these events, or he could end up mad that his brother is always along and gets your attention, plus little brother is going to want to participate and that may not be appropriate. Look into whether churches in your area do a "mothers' day out" program where you can drop off your younger child for short times on certain days; usually these programs then require you to volunteer as a sitter at other times to "pay" for your use of them. It would do your younger son good, too.
