The Robert Babcock of New Jersey is found in an 1830 census record at ancestry.com (if you click on his name in my answer above, it should take you to that record, but only if you have a subscription at ancestry.com). That same record is also here at familysearch.org. This record shows that Robert Babcock, of West Windsor Township in Middlesex County, NJ, was in his 20s and was living with a female in her 20s (presumably his wife), a girl age 5-9, and a boy under age 5 (presumably their two children).
As for your Vermont Babcocks, I have to admit I have no special knowledge of this family. My main research strategy, though, is to go beyond census records and vital records (which are commonly indexed and easily found on services like ancestry.com or familysearch.org) and also to look at land records (which are not indexed in these databases, and thus take a little bit more work to find), because (at least for farmers and others who owned land) the land records can give important clues about residence and family relationships.
In Vermont land records were kept at the town rather than county level. Land records for Wardsboro can be found at https://www.familysearch.org/search/catalog/140883. As I thumb through the index, the first mention of a Babcock is for two purchases by a David Babcock, recorded in vol. 3, pp. 425 and 506. The first of these is a purchase on 9 Oct 1806 by David Badcock [sic], then living in the town of Summersett (i.e., Somerset) in Windham County, VT, of two plots in the north district of Wardsboro totaling about 71 acres. The second deed records the purchase on 3 May 1808 by David Badcock, now of Wardsboro, of another 19 acres in the north district of Wardsboro.