Ortigia is small enough to navigate entirely on foot — in fact, a car is more trouble than it's worth on the island. The two main bridges from the mainland are the Ponte Umbertino and Ponte Santa Lucia, both pedestrian-friendly. The island's streets are narrow, often cobbled, and confusing to navigate without a map, but getting lost here is more pleasure than inconvenience.
The Neapolis Archaeological Park on the mainland is about a 20-minute walk from the Ortigia bridges, or you can take a taxi for a few euros. City buses also run between the park and the city centre. A bicycle is useful for reaching the Museo Paolo Orsi and other mainland sites, and rental is available on Ortigia.