Franeker has a small railway station on the line between Leeuwarden and Harlingen, with two trains per hour in each direction. From Leeuwarden the journey takes about fifteen minutes; from Amsterdam it's around two and a half hours by direct train via Leeuwarden.
By car, take the A31 motorway and exit at Franeker. There is free parking in several lots just outside the historic centre — the old town is small enough that you'll cover it all on foot. The town is also reachable by boat: a network of canals connects it to the rest of Friesland's Eleven Cities.