There is a bus that runs a couple times a day from Monterey directly to Big Sur
For hitching, it is best to hitch out of the Monterey Peninsula from the Barnyard Shopping Village which is located in Carmel Valley. This spot can be difficult to walk to and get rides in as it an affluent community. However, it is the last reasonably priced grocery opportunity before the exorbitant price-gouging that characterizes Big Sur and many tourists and travelers stop here before heading on. There is a pullout on the right shoulder directly after a set of 4 way traffic lights on the Highway 1 right in front of the shopping village. This is the best spot to hitch.
You can get to this spot by taking a Carmel/Carmel Valley bus from downtown Monterey.
North to Santa Cruz
South to Big Sur and Santa Barbara
East to Salinas
Getting into Monterey is easy from the North. An express bus leaves twice a day except weekends from downtown Santa Cruz for $3.00, or you can hitchhike from onramps in Santa Cruz. There is also a bus from Salinas.