To start at the beginning, everyone born on or after June 1st, 1975 must pass a state approved hunter safety course before you can even purchase a hunting license.
Once you have that, you can set up your account at gooutdoorsflorida.com, and purchase licenses and apply for WMA permits.
For small game hunting on a WMA for example, you will need a hunting license ($17) and a Management area permit ($26.50).
Generally for small game one doesn't use quota permits, (you have to apply for these months before, it's a random drawing for deer and turkey mostly) but check your specific WMA brochure to be on the safe side.
Most management areas will have a box of cards at the check station as you drive in, take one for each hunter in your vehicle, keep it on your person, and fill it out and put it back in the box when you leave.
As far as your pistol goes, as long as it's legal to hunt with at the time, you can carry it however you want.
I would think a charger would be easiest on a sling, but that's just me.
Most important of all, be sure to leave the squirrel's antlers and beards attached until you get home, otherwise you risk getting a ticket
Here's some links that might be helpful.
https://license.gooutdoorsflorida.com/Licensing/CustomerLookup
https://myfwc.com/hunting/wma-brochures/
Also the FWC has an app that's convenient, it's called,
Fish | Hunt FL
Hope some of this helps, Cheers!