You can reach Spetses either by boat from Piraeus port or by car via Kosta.
By boat: Hellenic Seaways Flying Dolphins depart on a daily basis from Piraeus and with frequent schedules to Spetses in 2-3 hours, depending on the itinerary. The ticket costs €77/person return.
By car: To get to Spetses by car, you will have to drive about 178 km from Athens towards Kosta in Argolida for about 2.5 hours. There, you will leave your vehicle at the parking lot (€5/day) and you will cross over to the island with sea taxis located at the port. Their capacity is up to 8 people and the ticket costs €20 (regardless of the persons).
Alternatively, you can reach Kosta with any of the KTEL Argolidas buses departing from Kifissos station in Athens.