Rhodes International “Diagoras” (RHO) sits about 14 km southwest of Rhodes Town at the northern tip of the island, and Lindos is a long way down the east coast from there, so the journey is three choices rather than one. A taxi or a pre-booked transfer is the arrival-day answer for most people going straight to Lindos, because it skips the town leg entirely. Book it rather than assuming a rank.
The KTEL bus is the cheap, dependable alternative, but it runs out of Rhodes Town rather than the airport, and the leg down to Lindos takes about 1 hour 40 minutes each way once you get there. A hire car only earns its keep if you want the west coast, the mountain wineries and the inland villages, which have little or no bus service. Whichever you pick, do not plan to drive into the Old Town: it is walled, car-free and walkable end to end.
Diagram by Greece Trips. Free to reuse with credit - CC BY 4.0.
The three ways, and what decides between them
The KTEL bus runs the east coast reliably, out of Rhodes Town rather than the airport - to Faliraki, Kalithea, Kolymbia and all the way south to Lindos. It is the cheap, dependable option and it is the backbone of getting around Rhodes without a car.
The catch is time. The KTEL bus down to Lindos takes about 1 hour 40 minutes each way from Rhodes Town, and you have to get from the airport into town first. On arrival day, with luggage, that is a long afternoon.
A taxi or pre-booked transfer is the arrival-day answer for most people going straight to Lindos, precisely because it skips the town leg. Book it rather than assuming a rank: on an island the taxi rank is small, and it does not grow in August.
A hire car is the option that changes the trip rather than just the transfer. You do not need a car for an Old-Town-and-east-coast trip, but you rent one the moment you want the west - the west coast, the mountain wineries and the inland villages have little or no bus service. If any of that is on your list, collecting the car at the airport makes this journey the first leg of the trip instead of a cost.
Do not plan to drive into the Old Town
Worth saying here because it catches people on the way back. The Old Town is walled, car-free and walkable end to end, and the gates are for people, not vehicles. Park outside and walk in.
How this fits the base decision
Lindos, the resort strips and the Old Town are three very different bases, and the transfer question follows from which you chose rather than the other way round. Where to stay in Rhodes makes that decision properly, and the Rhodes itinerary sequences it.
If you are basing in Lindos and not renting a car, budget the bus time into your day trips as well - the 1 hour 40 applies in both directions, and it is the reason a Lindos base makes Rhodes Town a day out rather than an evening.
Our Rhodes guide has the wider picture, and getting around Greece covers the KTEL network generally.