Palm Beach International Airport: This is the airport that makes sense: Palm Beach Shores is roughly 15–20 miles away. Take an Uber/Lyft or taxi directly to your hotel; allow 25–40 minutes depending on bridge and US-1 traffic, and roughly $30–55 before tip. A rental car is worthwhile only if you are planning several trips beyond the immediate coast. Palm Tran can get you toward downtown West Palm Beach (Route 44 is the airport link), but it is not a practical one-seat airport-to-Shores transfer.
Fort Lauderdale–Hollywood International Airport: Use Brightline north to West Palm Beach if flight price or timing makes FLL attractive, then take Uber/Lyft for the final 15–25 minutes to Palm Beach Shores. Budget about 1½–2 hours door to door after the airport-to-station connection, plus the rail fare and a roughly $20–35 ride from West Palm Beach. Renting a car at FLL only makes sense if you will use it throughout the trip; I-95 is not a relaxing arrival experience.
Miami International Airport: Choose MIA only for a notably better international fare or nonstop. The least bad public-transport route is Metrorail from the airport to connect with Brightline at MiamiCentral, then Brightline to West Palm Beach, followed by rideshare; realistically allow 2½–3½ hours door to door. A direct rideshare/car is usually 1½–2½ hours in ordinary conditions and can be much worse in traffic. It is not a sensible airport for a short Palm Beach Shores break.
West Palm Beach Brightline and Tri-Rail stations: West Palm Beach is the useful rail gateway, not Palm Beach Shores itself. Brightline is the fast option from Miami/Fort Lauderdale; Tri-Rail is the slower, cheaper regional alternative. From either West Palm Beach station, use Uber/Lyft or taxi for the final 15–25 minutes; there is no rail or frequent visitor-friendly shuttle into the Shores.