Old newspapers are such a valuable resource for any genealogist. You will be amazed at the type of information that you can find in a newspaper.
All the obvious ones, births, deaths, marriages, engagements may show up but there are also court information if you ancestors were involved in any type of dispute or criminal case. Land records may also show up in the newspapers.
One thing that is often overlooked is financial information. Our ancestors lived through difficult times and they may have had suits filed against them for debt or they may even have had to file for bankruptcy. While we may not want to find this sort of information, it may very well be a part of your history.
The advertisements in the newspaper can also be informative, they may include names and information.
The social pages can tell you a lot about what your ancestors were doing at any particular time.
You may find out that great grandma was visiting her sister in Simsbury or that her niece from New York had arrived for a visit.
You can also find announcements of ship arrivals. While I can't find my grandmother at Ellis Island, I can find a record in the New York newspaper announcing the arrival of the ship she and her family were on. Also, check the weather, it is important to place your ancestor in the context of the times they lived in.
Where to find newspapers is always problematic, not all of them have survived but if you are willing to pay there are websites that allow you to search them. On the local level, check the library in town. I spent many hours perusing the Penny Press (the predecessor of the Middletown Press) looking at references to my family. Godfrey Library also has copies if you are looking in Connecticut.
Newspapers are a valuable resource of the times our ancestors lived in and you may find documentation of events in their lives about which you had no previous knowledge and which are not documented anywhere else. It is certainly worth looking into and if nothing else you will learn a lot about the times and places where your ancestors lived.