Major sites -- London (Hackney, Stoke-Newington, 8 mi ENE of London center)Abney Park Cemetery, established on the grounds of the London residence of Sir Thomas and Lady Mary Abney, near the end of the senior line of the family.From what I understand, it's in pretty rough condition but still worth the visit, especially if you do some research beforehand and are going to be in London anyway.
Willesley, near Ashby-de-la-Zouche, WNW of London, family home for near 400 years.Although the manor hall was demolished in 1953, the family church/chapel and at least part of the gatehouse remain.It is currently a scout camp, and the grounds are a golf course.It has a listing on Wikipedia (https://secure.wikimedia.org/wikipedia/en/wiki/Willesley), although some of the historical information there is inaccurate vis-a-vis how the Hastings family came into possession of it, becoming henceforth the Abney-Hastings family (not blood relations).Ashby Castle was defended by the Abney and Hastings families for the Royalist side during the (English) Civil War and was quite a bother to the roundheads.
Measham -- just S of Ashby -- middle branch of the family held Measham Hall, a mile or so east of the village, for several generations starting in the 16th century.I think it may still be standing, but am not sure.This branch morphed into the Wooten-Abney family, not sure if they are really blood relatives but a Wooten-Abney sued to get Willesley back into the family after the death of the last Abney-Hastings baronet, indicating he thought he had the stronger claim...
Leicester, big city NNW of London -- the most junior branch of the family settled here around 1540ish, becoming prosperous merchants; several Abneys served as Lord Mayor.As far as I can tell, all American Abneys descended from this branch.Most Abneys that aren't buried at Willesley are interred at St. Mary's de Castro church.
-- the following is somewhat speculative, as there is some confusion between the d'aubigny and d'aubigne families (the former is norman, the latter breton, both may have anglicized to d'albini) and ultimately their exact relationship to the Abney surname; a number of geneologies have us as direct descendants --
If you want REALLY ancient Abney sites, there are a pair of ancient villages/crossroads named Hungerton and Wywell in Lincolnshire that were D'Albini family lands in the 13th century, as well as Belvoir Castle (the first one of the four that have occupied the site), held for a number of generations after the Conquest by the same family. The first Earl of Arundel (and lord of Arundel Castle) was also of this lineage.
Hope that is helpful.With the exception of the ancient sites, that would be my ideal itinerary.