The local churches, the Warwickshire Parish Clerks and in particular the Warwick County Records Office are all good sources for anyone tracing their family history in this area.
Not all church memorials and gravestones are fully recorded online, so you may want to look around the churches in the parish. Many early memorials that are inside the church are more readable than those outside on graves. Don’t forget to look down, as many will be set into the floor and may be partly covered by church furniture. The Edkins are a local family with a long history in the parish. The memorial stone to Thomas and Elizabeth Edkins shown here is inside Aston Cantlow church. An Edkins family tree is available at this link.
A list of around 500 known graves in St John the Baptist Church in Aston Cantlow is provided here as a PDF file – Aston Cantlow Graves. You can open it with Abobe Acrobat or other PDF file reader and search for specific names in it. Maps of the churchyard showing the keys to the grave locations can be found in the St John the Baptist Church pages.
A good free starting point for investigating your family history is the International Genealogical Index compiled with data captured from parish records by the Church of Latter Day Saints.
All of the main genealogy websites will have copies of census entries for the parish from 1841 through to 1911 as well as many other sources such as indentures, land agreements, court actions etc, but you will usually need to pay a fee for retrieving anything other than basic details.
The UK Census Online is one of many that will give you a free search if all you have is a name and a date of birth or death, but will only provide limited information on the results before you have to pay.
Genuki’s Warwickshire section is also a good starting point, as it is completely free, and will point you at many sources relevant to families in Warwickshire and elsewhere.
As you go further back in your research before the 1841 census records it gets harder, and you will need to resport more to parish records of births, baptisms, marriages & deaths. These records were officially required to be kept after an injunction was introduced by Thomas Cromwell in 1538, but many parishes did not really start doing the job properly until nearly the end of the 1500s, which is probably why we don’t have an official record of the marriage of Shakespeares parents in Aston Cantlow Church in 1557. Many of these early records were lost in the English civil war, and quite a few of the later ones were lost to later wars, fire, flood or decay, but a surprising amount do exist and are available through your local records office in Warwick. You will be very lucky to get back before 1537 as you will need to try to find old manorial and land records unless it is a very prominent family with titles or coats of arms that have been recorded by the College of Arms. Good Luck.