As the name may suggest, Plymouth County was one of the original counties from Plymouth Colony, and was established in 1685. If you are looking for copies of Plymouth County public records, there is a good chance you will be able to find material that goes back at least this far, if you know where to look.
Modern records that go as far back as 1921 are kept by the state. They can be accessed with the proper forms from the Registry of Vital Records and Statistics. The office is located in Dorchester but you can send your application in by mail (a month's wait time) if you can't visit the office in person (while you wait service).
After 1921, the records are moved to the State Archives in Boston. There they can be researched by the public, and they are open during the usual business hours. You can research in person, or do more browsing through their holdings via the online search features of their website. Records are free to look through, though you will have to pay for any photocopies you make. Their collection dates to 1841, when the state first made it a requirement to file records registrations with the state government.
And all material from before 1841 has remained with the counties, and that is where you will have to put in your request. The county clerk's office in the town of Plymouth is the best place to start as they often have the records for the county, but you may also have to contact the town clerk where the even took place if the county office doesn't have the record you are looking for.
The forms for any of these requests can be picked up at the counter, or downloaded at home from the Office of Health and Human Services website. There is one form for all vital records.
Plymouth County Birth Records
There are fees associated with getting a certified copy of a Plymouth County birth record, but it will depend on how you make your request application. Mailed in forms need a $28 fee with them, but if you are at the counter, it will only cost you $18. There are no restrictions on who can order copies of records.
Plymouth County Death Records
Plymouth County death records cost the same as the birth records, and are also available to anyone who makes a request. There are no limitations based on your relationship to the deceased.
Plymouth County Marriage Records
You can also get a Plymouth County marriage record without restriction as well. They are available through the same offices and are issued for the same fees.
Plymouth County Criminal Records
Plymouth County criminal records are not classed as "vital records" and they are handled directly by the state, not the county. In particular, it's the Department of Criminal Justice Information Services that handles these documents. Your requests must go to them. The DCJIS website explains how to apply for documents, and which Plymouth County criminal records are considered public domain. The fee for these records is $30 for each one ordered.