Sarasota-Bradenton International Airport (SRQ) is the closest airport, with direct flights to several US cities including New York, Chicago, and Atlanta. It's smaller and more convenient than Tampa's airport, but has fewer routes. Tampa International Airport (TPA), about an hour north, has a much wider selection of flights and is worth checking for price comparison.
By car, Sarasota sits on I-75 about 60 miles south of Tampa. From the airport, the drive into downtown takes around 15 minutes. No practical train service reaches Sarasota — Amtrak's nearest station is Tampa, from where you'd need a rental car or bus.
Greyhound and Flixbus run services between Sarasota and Tampa, Miami, and other Florida cities. The Sarasota bus terminal is downtown on Fruitville Road.