After spending hours on the flight, the last thing you want to do is travel even longer in a car and deal with traffic. Especially Tampa traffic. Which is why I always suggest flying into the Sarasota-Bradenton International Airport if you can. Unlike Tampa International Airport, the Sarasota Airport is only a 30 minute drive away from AMI. With TPA being over an hour away.
While this airport may be closer, it unfortunately does not offer as many airlines/destinations as TPA does. Unlike Tampa which has over 60 non stop flights (including international flights), Sarasota-Bradenton only has 20, with none being international. However, they do appear to be slowly adding more destinations, so hopefully one day there will be more options. As for now, if you can get a flight to this airport rather than Tampa, I would highly suggest it.