At reply #1 Tunji recommends a Y-DNA test which can only be taken by males, and traces the male line only.
Ancestry only offer an autosomal test. If you wanted to go for a Y test, FTDNA would be the company to go with.
The trouble with the Y test is that any matches you do get may not be within a traceable time period. They are also substantially more expensive. I have not had any useful matches for the two Y tests I have experience of. They come into their own if you have two males who wish to prove that they have a descendant in common.
The minus side of an autosomal test is that you will get a lot of matches but you probably won’t know if the matches are from your Connolly side or the other side of your family. If you have an older Connolly relative it might be helpful to have them tested too, so you can look for common matches.
Also consider matches with different surnames as female Connollys who married are likely to have changed their surnames.
Everyone’s results are different so whether a DNA test will help you find your Connolly origins or not is unknown. It is fairly certain that you will get some decent results on other branches though.
Good luck.
