In order to do cemetery research you will first need to learn where your ancestor is buried. Here are a few ways to do that:
Locate death records and/or obituaries. They nearly always mention the cemetery.
Ask family members if they have any funeral, mass or prayer cards. Also ask if they know of family burial locations.
Funeral homes can be helpful in locating cemetery records. The funeral home is often mentioned in obituaries and death notices. Also, funeral directors will have knowledge of the cemeteries in their area.
Check with the church your ancestors attended. Often a record of the death and burial is kept by the church. Also, some churches have attached cemeteries and will have records of the burials there. If you don't know the church, check a map of the area where your ancestor lived and see what churches are near where they lived.
Don't forget to check with the local genealogical or historical society. Quite often they have compiled cemetery indexes.
On-line sources can be very helpful and there are quite a few. Check out Cindi's List at http://www.cyndislist.com/deaths.htm for a varied list.