I grew up in Tarpon Springs (lived there for 17 years). I always went through Tampa, it was always the easiest way. I take US 19 South to St rd 60 (Gulf to Bay) East. Follow St rd 60 all the way across Tampa Bay (you cross over the Courtney Campbell Causeway and enter into Tampa) follow the signs to Tampa Airport (the exit is just after you come off the causeway bridge maybe about a mile or two) once you get onto Memorial Highway (the exit for the airport off St Rd 60) move over one lane (the right lane forces you off to the airport.) Follow that to the I-275 exchange and get on I-275 East that will bring you through Tampa and connect to I-4 once on I-4 the Disney exits start at exit 62 (I think, I know the Epcot/ DTD exit is 67).
Going up to SR 54 works too. Might be easier right now because of all the construction on US 19 through Countryside and Clearwater.