The real taxis in Lindos (as on most of Rhodes) are modern air-con equipped Mercedes. The taxis depart from near the main square, down the hill from the bus stop. If you want to stay late in Lindos to see it by night, you'll need to get a taxi back to Rhodes (or wherever you;re staying). Busses don't run much later than 6 or 7.
The cost from Lindos to Rhodes Town is 25 euro. You can haggle with the driver, especially if you catch a Rhodes Town taxi that is gonna have to go back empty otherwise. You can get a rate of 3 euro per person this way, which compares with the bus, and is cooler and quicker.
The walk up to the Acropolis is quite steep and long, and on a hot, hot day in August it might prove beyond the elderly or unfit. Even if you ARE fit (or young) you might still like to go up on a donkey.
The donkeys run between the main street at the edge of the village, where you enter the shady, narrow streets from the main open square.