If you're coming from Newcastle, the fastest way to get to South Shields is to take the A19 south for about 10 miles, then take the A194(M) towards South Shields. Alternatively, you can take the Tyne Tunnel if you don't mind paying a toll.
If you're coming from Sunderland, take the A183 east towards South Shields.
If you're traveling by train, South Shields has its own station on the Tyne and Wear Metro system. You can take the Metro from Newcastle or Sunderland to get there.
If you're looking to sightsee on the way, consider taking the scenic route along the A1300 coast road from Newcastle, or the route along the River Wear from Sunderland.
Overall, driving or taking the Metro are likely your best options for getting to South Shields, but there are plenty of routes to choose from depending on where you're starting from and what you want to see along the way.