If you live in Ct, and own rental property in Fla, if you sell the rental property in Fla., does CT charge a capital gains tax? Or do they only charge if you sell a property in CT?

All of your income is subject to tax by the state that you live in, no matter where the income is from. So you have to report the sale of the Florida property on your CT tax return and you will have to pay CT tax on the capital gain from the sale.