The closest major city to the Eyre Peninsula is Adelaide. You have the option of taking a plane, bus, ferry or driving to get to the Eyre Peninsula.
Commercial flights operate to the Eyre Peninsula towns of Port Augusta, Whyalla, Port Lincoln and Ceduna from Adelaide, or from the Yorke Peninsula west of Adelaide. These are run by Regional Express and Qantaslink. Because of the geography of the area, flights are often short and quick, and offer a practical way to access the area at around $150.
The Eyre Peninsula towns are connected by the Eyre Highway across the inland edge, the Flinders Highway along the west coast and Lincoln Highway along the east coasts. Port Augusta, the Peninsula’s easternmost town, is around four hours drive from Adelaide at the beginning of the Lincoln Highway.
The Lincoln Highway is the one you'll probably use the most, it runs between Port Augusta and Port Lincoln down the east coast of the peninsula. The road is sealed and good quality with a speed limit of 110 km/h. There are no overtaking lanes for its entire distance, but there are long straight sections that allow for safe overtaking in good visibility conditions. It's used by oversized road trains (long and large trucks!), which can test your patience when travelling behind them and thinking of overtaking. From Port Augusta to Whyalla the road travels away from the coast, and the terrain becomes arid. There are no real stopping off points on this section. From Whyalla the road continues south onto Port Lincoln, once more travels closer to the coast and you begin to encounter regular towns where you might want to stop and visit. Each of these comes with mandatory piers, lovely beaches, bushwalks, accommodation and food. Stops with good facilities are Cowell, Arno Bay, Port Neill, and Tumby Bay - each less than an hour apart.
Premier Stateliner runs a daily service between Port Lincoln and Adelaide, with a second service terminating at Whyalla. It stops at Cowell, Arno Bay, Cleve, Port Neill, and Tumby Bay en route. They also run several services a week to Ceduna, via Iron Knob, Kimba and Streaky Bay (best names ever!) You can view the timetables and book tickets from their website.
Sea SA Car & Passenger Ferries operate a ferry service across the Spencer Gulf between Wallaroo on the Yorke Peninsula and Lucky Bay (near Cowell) on the Eyre Peninsula. It can carry both cars and passengers and takes just over two hours.