There are many places you can go for evidence to prove the kinship links in your Mayflower lineage application. Here on our site you can consult the Research pull-down menu above for appropriate books and articles, great libraries, professionals you can hire, and scholarly journals we recommend. And now, below, see the resources we've brought over from our previous website. Indices to Berkshire and Franklin County Probates, among other. We will slowly be updating the information we provide here, so stay tuned.
For detailed information about each state, consult Michael J. Leclerc, Genealogist's Handbook for New England Research, 5th ed. (Boston: New England Historic Genealogical Society, 2012). It is available in most local libraries.
A great overall introduction to any locality or record group can now be found on the FamilySearch website in their Wiki. Simply enter the state you are researching in the search box. Some places have a more complete description than others, but check back often, as they are constantly being updated. You can find a list of all US collections on their site, browsable or indexed, here.