The old town is compact and easy on foot. From the train station to the cathedral is about a 20-minute walk along the Maximilianstrasse, Speyer's main axis, and everything in the historic centre — the Altpörtel, the churches, the Mikwe, the museum — is within a few minutes of that route. You do not need any transport to cover the centre; a comfortable morning or afternoon is enough to walk it end to end.
The Technik Museum is an exception. It lies about a kilometre south of the old town — around 20 minutes on foot, or a short taxi or bus ride. If you are combining the cathedral and museum in a single day, the walk between them is straightforward but not particularly interesting, so a taxi or the local bus (route 564) is worth considering for at least one direction, especially in summer heat or with children in tow.