PRONI has a more or less full set of copies of the Dublin registry of Deeds (the odd reel is missing). That covers all counties north and south from the start (1709) up to around 1930.
You can search under townland or under the grantor’s name. Townland is generally easier, especially in the early years, when the names index doesn’t say where in Ireland the person lived. (They included that from around 1830 onwards). Also if you don’t know who the grantor was, then the names index is a waste of time.
It takes a little while to understand how the Deeds records are kept. You use one set of films to find possible deeds, and then you have to order a different film with the memorial (summary of the deed) on it. The handwriting can sometimes be tricky to follow. Allow plenty of time.
Access to the PRONI copies is free, whereas I think there may be a fee to pay if you go to the actual Registry of Deeds in Dublin and view them there. (You would need to check on that).
The Registry of Deeds is separate to the Land Registry. PRONI only has copies of the Registry of Deeds records.
Answering your second point about how did they accumulate money to buy land, are you sure they actually owned it? What proof of ownership do you have? Statistically it is far more likely they rented it, and just paid next year’s rent out of this year’s profits.
If you do think they owned land (1 acre or more) in 1876, then they should appear on the landowners list. If they are not there then they were probably renting. See:http://www.failteromhat.com/lo1876.php