The easiest way to go from the harbor at Marina Grande to Capri city center is by funicular. This small train takes you in 3 minutes to Piazza Umberto I in the center of the city. The station is located in the west of the harbor. There'ss at least every 1 train every 10 minutes and you pay € 1.80 per person (one way).
Despite its small size, there are good bus connections on the island. Every 15 to 20 minutes you can take a bus from Marina Grande to Capri center. The bus takes you in 10 minutes to Piazza Martiri d'Ungheria for € 1.80 per person (one way).
From down at the harbor at Marina Grande, you can walk to the upper town of Capri. The road is steep, but you will be rewarde with some breathtaking views. Depending on your condition it takes 25-45 minutes. You can start your hike near the ferry station.
In the harbor you can take a taxi to the center of Capri for € 15 per taxi (4 persons one way).
A bus ride to the 5 km far away Anacapri takes 25 minutes and costs € 1,80 per person (one way). Anacapri, the bus stop in Piazza della Pace.
TIP: If you plan to make several bus rides, you can also buy a day pass for € 8.60 per person.
A taxi to Anacapri takes about 20 minutes and costs € 20 per taxi (4 persons one way).
From Marina Grande you can make an excursion by boat to the Grotto Azura. The boats leave from the ferry station and costs € 15 per person (return, excluding entrance fee of € 13).
Once at the caves you step into smaller rowing boats. The entrance to the cave is very low so you must lie almost down. In wild sea or storm the tour is cancelled.