The Florence airport is not that far from the city center but there is no direct rail connection to the historical center.
You have two options, either take a taxi to your hotel or take the public shuttle bus to the main train station of Santa Maria Novella. If you know your hotel is just a block or two from the train station, then the cheapest option will be to take the shuttle and walk the rest of the way to your hotel.
The bus shuttle from the airport is called "VolainBus" and the stop where you get on is outside not far from the exit as you leave the baggage and customs area in the airport (although it isn't visible). The shuttle bus goes directly to the Santa Maria Novella train station without stops and costs 4 euros (buy the ticket from the shuttle driver).