An Ancestry test would be a good starting point simply as they have by far the biggest tester database.
What the test results will show is a long list of DNA Matches, that is people who share DNA with in your case you GGM, the match list will be listed in terms of highest cM going down to lowest cM. The cM stands for centimorgan and it is the measure of how much DNA you share. As an Example a DNA match who shares 100cM is likely to be about a Third Cousin.
It will also show the Ethnicity Estimate, which may or may not be accurate. It is a WIP by Ancestry and by any other testing company.
A y-DNA test can only be taken by a male as the Male y chromosome is only passed Male to Male hence it just goes down the Paternal line.
There is a Female sort of equivalent the mtDNA test that follows the Female to Female line, Mother passes to Daughter who passes to her Daughter etc. A Mother passes the mtDNA gene to both her Daughter and to her Son but the Son will not pass it on to his children hence a Male can also take a mtDNA test.
DNA like the y-DNA goes back to the beginning of our race as does mtDNA, but practically the combined Autosomal DNA test that Ancestry uses goes back centuries. There is a but in that a match below 10cM can be inaccurate.
There is a website called DNA Painter, enter the cM value of a DNA Match into its Shared tool and the results will show the percentage probability of a set of specific relationships.
Read this stickie thread.
https://www.rootschat.com/forum/index.php?topic=863488.0