The best place initially would be PRONI and to use the available Church records. However if you are not able to attend PRONI - (Public Record Office Northern Ireland) in person, thats not much use to you at present.

Take a look at the County Pages link on the Home Screen for this site. This will give you a good background into what records are available and just about all of them are free to use. Be aware that Belfast straddles Counties Antrim & County Down!!

Have a look at the website John Grenhams names - it will give you some insight into the distribution of Surnames. John KNOWS about Irish family History

There will be no civil records covering the time period you mention These are held at GRONI (General Record Office N.I.) and will cost to view. But they dont start until 1845 for marriages and 1864 for births and deaths. Not much use in your case. Also these same records are available for free up to 1922 on Irish Genealogy.

https://civilrecords.irishgenealogy.ie/churchrecords/civil-search.jsp

However the deaths may include relatives and are worth perusing.

Knowing what religion your forebears were will help greatly in knowing where to search. Catholic Parish records are online and are mostly transcribed on Ancestry, Find my past and FamilySearch. Church of Ireland records were badly damaged in 1922 in Dublin and many were lost completely.

There are no census records until 1901, then 1911. Not much use.

Have a really good look on the County Pages (Antrim & Down) on the group Home Screen so as you can devise a good search strategy,

Dont worry about perceived lack of experience. We're all learning. If /when you get stuck come back and ask questions. We all learn that way.