Murphys is not reachable by public transit. A car is required, and that's not going to change — the town sits in the Sierra Nevada foothills with no rail connection and limited bus service to the surrounding area.
From the Bay Area, take I-580 or I-205 east to Stockton, then Highway 4 east through Angels Camp. From Angels Camp it's about 30 minutes on Highway 4 east directly into Murphys. The total drive from the Bay Bridge is roughly 2.5 hours in normal traffic. From Sacramento, take Highway 99 south to Stockton, or Highway 50 east to connect to Highway 49 south through Jackson to Angels Camp, then Highway 4 east into Murphys. The Sacramento drive is typically 2 hours or less. Angels Camp makes a natural fuel and snack stop in either direction and is worth a brief detour in its own right — the Calaveras County seat has a few things worth seeing.