Not all cemeteries are privately run, some are abandoned, some are run by the town, some by the government. Each has their own set of rules and what they share with the public. Many will freely provide all their records on your ancestor, others charge a steep fee (like Rock Creek Cemetery in Washington, DC). Often you can find a volunteer to do some of the leg work for you. In the past, if I received something from them, I always wrote a thank-you and enclosed a small token of $10. Always pay them in some way, either dollars or stamps, because you never know if you will need their service again, and they will remember you. If their work is exceptional and they are employed, write a thank you letter, so it can go in the employee's human resources file (I did that once and even brought the administrative assistant a box of chocolates.)
The next time I need information; I will either call or email a request. Personally, I like calling best. Do it mid day, never just before closing. I'll explain exactly what I am looking for, get right to the point. Sometimes I'll have a check-list, so I don't forget to ask something. Often on the phone, I get a little nervous or side tracked, then the call isn't completed in full. I would still do that, especially if it is a Historical Society (those people can be so nice)
A great source for locating historical societies is the book, The Genealogist's Address Book by Elizabeth P. Bentley. The new 6th edition was published in 2009 and has 809 pages. I used to own a copy, and referred to it frequently, mine didn't even have email addresses, it was that old. A page snippet from the new book is below.
My ancestor is # 59
Example of a pamphlet.